A TEXTBOOK OF

ENGINEERINGMATHEMATICS

For

B.Sc. (Engg.), B.E., B. Tech., M.E. andEquivalent Professional Examinations

By

N.P. BALI Dr. MANISH GOYALFormerly Principal M.Sc. (Mathematics), Ph.D., CSIR-NET

S.B. College, Gurgaon Associate Professor

Haryana Department of Mathematics

Institute of Applied Sciences & Humanities

G.L.A. University, Mathura,

U.P.

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CONTENTS

1. Complex Numbers .......................................................................................... 1–831.1. Real Numbers ........................................................................................................ 11.2. Basic Properties of Real Numbers ........................................................................ 11.3. Complex Numbers ................................................................................................. 21.4. Conjugate Complex Numbers ............................................................................... 21.5. Geometrical Representation of Complex Numbers ............................................. 21.6. Properties of Complex Numbers ........................................................................... 31.7. Standard Form of a Complex Number ................................................................. 31.8. Effect of Rotation, in the Anti-clockwise Direction, Through an Angle

on the Complex Number ..................................................................................... 121.9. De Moivre’s Theorem .......................................................................................... 201.10. Roots of a Complex Number ............................................................................... 301.11. Exponential Function of a Complex Variable .................................................... 531.12. Circular Functions of a Complex Variable ......................................................... 541.13. Trigonometrical Formulae for Complex Quantities .......................................... 551.14. Logarithms of Complex Numbers ....................................................................... 571.15. The General Exponential Function .................................................................... 601.16. Hyperbolic Functions .......................................................................................... 631.17. Formulae of Hyperbolic Functions ..................................................................... 651.18. Inverse Hyperbolic Functions ............................................................................. 721.19. C + iS Method of Summation .............................................................................. 75

2. Theory of Equations and Curve Fitting ..................................................... 84–1382.1. Polynomial ........................................................................................................... 842.2. Zero Polynomial ................................................................................................... 842.3. Equality of Two Polynomials .............................................................................. 842.4. Complete and Incomplete Polynomials .............................................................. 842.5. Zero of a Polynomial ............................................................................................ 852.6. Division Algorithm .............................................................................................. 852.7. Polynomial Equation ........................................................................................... 852.8. Root of an Equation ............................................................................................. 852.9. Synthetic Division ............................................................................................... 862.10. Fundamental Theorem of Algebra...................................................................... 882.11. Multiplication of Roots ........................................................................................ 932.12. Diminishing and Increasing the Roots ............................................................... 942.13. Removal of Terms ................................................................................................ 96

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2.14. Reciprocal Equations ........................................................................................ 1002.15. Sum of the Integral Powers of the Roots and Symmetric Functions .............. 1052.16. Symmetric Functions of the Roots .................................................................... 1092.17. Descarte’s Rule of Signs .................................................................................... 1112.18. Cardon’s Method .............................................................................................. 1112.19. Irreducible Case of Cardon’s Solution .............................................................. 1162.20. Descarte’s Method ............................................................................................. 1172.21. Ferrari’s Solution of the Biquadratic ............................................................... 1202.22. Curve Fitting ..................................................................................................... 1222.23. Graphical Method .............................................................................................. 1222.24. Method of Group Averages ............................................................................... 1242.25. Equations Involving Three Constants .............................................................. 1262.26. Principle of Least Squares ................................................................................ 1302.27. Method of Moments ........................................................................................... 136

3. Matrices ...................................................................................................... 139–1943.1. Definitions (Matrices) ........................................................................................ 1393.2. Addition of Matrices .......................................................................................... 1423.3. Multiplication of a Matrix by a Scalar ............................................................. 1423.4. Properties of Matrix Addition ........................................................................... 1433.5. Matrix Multiplication ........................................................................................ 1443.6. Properties of Matrix Multiplication.................................................................. 1463.7. Transpose of a Matrix ....................................................................................... 1493.8. Properties of Transpose of a Matrix ................................................................. 1493.9. Symmetric Matrix ............................................................................................. 1503.10. Skew-symmetric Matrix (or Anti-symmetric Matrix) ...................................... 1503.11. Every Square Matrix can Uniquely be Expressed as the Sum of a

Symmetric Matrix and a Skew-symmetric Matrix .......................................... 1513.12. Orthogonal Matrix............................................................................................. 1513.13. For any Two Orthogonal Matrices A and B, Show that AB is an

Orthogonal Matrix............................................................................................. 1513.14. Adjoint of a Square Matrix ............................................................................... 1523.15. Singular and Non-singular Matrices ................................................................ 1533.16. Inverse (or Reciprocal) of a Square Matrix ...................................................... 1533.17. The Inverse of a Square Matrix, if it Exists, is Unique ................................... 1533.18. Theorem : The Necessary and Sufficient Condition for a Square Matrix

A to Possess Inverse is that | A | 0 (i.e., A is Non-singular) ....................... 1533.19. If A is Invertible, Then so is A–1 and (A–1)–1 = A ............................................... 1553.20. If A and B be Two Non-singular Square Matrices of the Same Order,

then (AB)–1 = B–1 A–1 .......................................................................................... 1553.21. If A is a Non-singular Square Matrix, then so is A and (A )–1 = (A–1) ............ 1553.22. If A and B are Two Non-singular Square Matrices of the Same Order, then

adj(AB) = (adj B) (adj A) .................................................................................... 156

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3.23. Elementary Transformations (or Operations) .................................................. 1573.24. Elementary Matrices ......................................................................................... 1583.25. The Following Theorems on the Effect of E-operations on Matrices

Hold Good .......................................................................................................... 1583.26. Inverse of Matrix by E-operations (Gauss-jordan Method) ............................. 1593.27. Rank of a Matrix ............................................................................................... 1603.28. Solution of a System of Linear Equations ........................................................ 1653.29. Vectors ............................................................................................................... 1713.30. Linear Dependence and Linear Independence of Vectors ............................... 1713.31. Linear Transformations .................................................................................... 1723.32. Orthogonal Transformation .............................................................................. 1733.33. Complex Matrices .............................................................................................. 1753.34. Characteristic Equation .................................................................................... 1783.35. Eigen Vectors ..................................................................................................... 1783.36. Cayley Hamilton Theorem ................................................................................ 1813.37. Reduction of a Matrix to Diagonal Form.......................................................... 1843.38. Quadratic Forms ............................................................................................... 1863.39. Linear Transformation of a Quadratic Form ................................................... 1873.40. Canonical Form ................................................................................................. 1873.41. Index and Signature of the Quadratic Form.................................................... 1883.42. Definite, Semi-definite and Indefinite Real Quadratic Forms ........................ 1883.43. Law-of-inertia of Quadratic Form .................................................................... 1883.44. Reduction to Canonical Form by Orthogonal Transformation ........................ 191

4. Analytical Solid Geometry ........................................................................ 195–3364.1. Introduction ....................................................................................................... 1954.2. Co-ordinate Axes and Co-ordinate Planes ....................................................... 1954.3. Co-ordinates of a Point ...................................................................................... 1954.4. Distance between Two Points ........................................................................... 1974.5. Section Formula ................................................................................................ 1984.6. Centroid of a Triangle ....................................................................................... 2014.7. Tetrahedron ....................................................................................................... 2014.8. Centroid of a Tetrahedron ................................................................................ 2024.9. Angle between Two Skew (or Non-coplanar) Lines ......................................... 2034.10. Direction Cosines of a Line ............................................................................... 2034.11. A Useful Result ................................................................................................. 2034.12. Relation between Direction Cosines ................................................................. 2044.13. Direction Ratios of a Line ................................................................................. 2054.14. Direction Ratios of the Line Joining Two Points ............................................. 2064.15. Angle between Two Lines ................................................................................. 2064.16. Find the Angle between Two Lines whose Direction Ratios are a1, b1, c1

and a2, b2, c2. Deduce the Condition for Perpendicularity and Parallelismof Two Lines ....................................................................................................... 208

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4.17. Projection ........................................................................................................... 2164.18. To Prove that the Projection of the Join of two Points (x1, y1, z1), (x2, y2, z2)

on a Line whose Direction Cosines are l, m, n is l(x2 – x1) + m(y2 – y1)+ n(z2 – z1) .......................................................................................................... 216

4.19. The Plane ........................................................................................................... 2184.20. General Equation of First Degree in x, y, z Represents

a Plane ............................................................................................................... 2184.21. Intercept Form .................................................................................................. 2194.22. Normal Form ..................................................................................................... 2214.23. Three Point Form .............................................................................................. 2234.24. (a) Angle between Two Planes .......................................................................... 2254.24. (b) Perpendicular Distance of a Point from a Plane ........................................ 2274.25. Any Plane Through the Intersection of Two Given Planes ............................. 2294.26. Planes Bisecting the Angles between Two Planes ........................................... 2314.27. Projection on a Plane......................................................................................... 2324.28. Theorem ............................................................................................................. 2324.29. General Form .................................................................................................... 2374.30. Symmetrical Form............................................................................................. 2374.31. Reduction of the General Equations to the Symmetrical Form ...................... 2414.32. Perpendicular Distance Formula ..................................................................... 242

4.33. To Find the Point of Intersection of the Line x xl

y ym

z zn

1 1 1

with the plane ax + by + cz + d = 0 ................................................................... 248

4.34. The Conditions that the Line x xl

y ym

z zn

1 1 1 may be Parallel to

the Plane ax + by + cz + d = 0 are al + bm + cn = 0 andax1 + by1 + cz1 + d 0 ......................................................................................... 249

4.35. The Conditions that the Line x x

ly y

mz z

n1 1 1 may Lie in the Plane

ax + by + cz + d = 0 are al + bm + cn = 0 and ax1 + by1 + cz1 + d = 0 ............... 249

4.36. The Condition for the Line x x

ly y

mz z

n1 1 1 to be Perpendicular

to the Plane ax + by + cz + d = 0 ....................................................................... 2494.37. Angle between a Line and a Plane ................................................................... 2534.38. Any Plane Through a Given Line ..................................................................... 253

4.39. To Find the Condition that the Two Lines x xl

y ym

z zn

1

1

1

1

1

1

,

x xl

y ym

2

2

2

2

= z z

n2

2 may Intersect (or May be Coplanar)

and to Find the Equation of the Plane in which they Lie ............................... 2614.40. Shortest Distance between Two Lines ............................................................. 265

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4.41. Magnitude and Equations of Shortest Distance .............................................. 2654.42. Intersection of Three Planes ............................................................................. 2754.43. Definition (The Sphere) ..................................................................................... 2814.44. Equations of a Sphere in Different Forms ....................................................... 2814.45. Touching Spheres .............................................................................................. 2824.46. Four-point Form ................................................................................................ 2834.47. Diameter Form .................................................................................................. 2844.48. Section of a Sphere by a Plane.......................................................................... 2894.49. Intersection of Two Spheres ............................................................................. 2904.50. Equations of a Circle ......................................................................................... 2904.51. Any Sphere Through a Given Circle ................................................................. 2944.52. Great Circle ....................................................................................................... 2944.53. Definition of the Tangent Plane ....................................................................... 2984.54. Equation of the Tangent Plane at a Point........................................................ 2984.55. Angle of Intersection of Two Spheres ............................................................... 3034.56. Condition of Orthogonality of Two Spheres ..................................................... 3044.57. Definition (The Cone) ........................................................................................ 3084.58. Equation of the Cone with Vertex at the Origin .............................................. 3084.59. The Direction Cosines (or Direction Ratios) of a Generator of a

Cone Satisfy the Equation of the Cone whose Vertex is the Origin ............... 3114.60. Quadric Cone Through the Axes ...................................................................... 3114.61. Right Circular Cone .......................................................................................... 3124.62. To Find the Equation to the Cone whose Vertex is the Point ( , , ) and

Base the Conic F(x, y) = ax2 + by2 + 2hxy + 2fy + 2gx + c = 0, z = 0 ................. 3154.63. Enveloping Cone ................................................................................................ 3174.64. Angle between Two Lines in which a Plane Through the Vertex Cuts

a Cone ................................................................................................................ 3184.65. Definitions (The Cylinder) ................................................................................ 3234.66. To Find the Equation to the Cylinder whose Generators are Parallel

to the Line xl

ym

zn

and Intersect the Curve .............................................. 324

4.67. Equation of Right Circular Cylinder ................................................................ 3264.68. Enveloping Cylinder .......................................................................................... 3284.69. Definition (The Conicoids) ................................................................................ 330

5. Succesive and Partial Differentiation ...................................................... 337–4265.1. Successive Differentiation ................................................................................ 3375.2. Calculation of nth Order Derivatives ................................................................ 3375.3. Use of Partial Fractions .................................................................................... 3425.4. Leibnitz Theorem .............................................................................................. 3455.5. Determination of the Value of The nth Derivative of a Function at x = 0 ....... 3515.6. Function of Two Variables ................................................................................ 354

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5.7. Continuity .......................................................................................................... 3545.8. Partial Derivatives of First Order .................................................................... 3555.9. Partial Derivatives of Higher Order ................................................................. 3565.10. Homogeneous Functions ................................................................................... 3635.11. Euler’s Theorem on Homogeneous Functions .................................................. 3645.12. If u is a Homogeneous Function of Degree n in x and y, ................................. 3645.13. Deductions From Euler’s Theorem ................................................................... 3655.14. Composite Functions ......................................................................................... 3725.15. Differentiation of Composite Functions ........................................................... 3735.16. Taylor’s Theorem for a Function of Two Variables .......................................... 3805.17. Jacobians ........................................................................................................... 3855.18. Definitions ......................................................................................................... 3855.19. Properties of Jacobians (Chain Rules) ............................................................. 3855.20. Theorem ............................................................................................................. 3865.21. Jacobian of Implicit Functions ......................................................................... 3875.22. Functional Relationship .................................................................................... 3885.23. Approximation of Errors ................................................................................... 3975.24. Maxima and Minima of Functions of Two Variables ....................................... 4035.25. Conditions for F(x, y) to be Maximum or Minimum ........................................ 4045.26. Rule to Find The Extreme Values of a Function z = f(x, y) ............................. 4045.27. Conditions for f(x, y, z) to be Maximum or Minimum ...................................... 4055.28. Lagrange’s Method of Undetermined Multipliers ........................................... 4085.29. Geometrical Meaning of Partial Derivatives ................................................... 4175.30. Tangent Plane and Normal to a Surface .......................................................... 4185.31. Differentiation under Integral Sign ................................................................. 420

6. Multiple Integrals ....................................................................................... 427–4756.1. Double Integrals ................................................................................................ 4276.2. Evaluation of Double Integrals ......................................................................... 4286.3. Evaluation of Double Integrals in Polar Co-ordinates .................................... 4346.4. Change of Order of Integration ........................................................................ 4376.5. Triple Integrals ................................................................................................. 4406.6. Change of Variables .......................................................................................... 4426.7. Area by Double Integration .............................................................................. 4496.8. Volume as a Double Integral ............................................................................ 4496.9. Volume as a Triple Integral .............................................................................. 4556.10. Volumes of Solids of Revolution........................................................................ 4576.11. Calculation of Mass ........................................................................................... 4586.12. Centre of Gravity (c.g.) ...................................................................................... 4606.13. Centre of Pressure ............................................................................................. 4636.14. Moment of Inertia ............................................................................................. 466

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6.15. Product of Inertia .............................................................................................. 4676.16. Principal Axes.................................................................................................... 467

7. Vector Calculus ......................................................................................... 476–5327.1. Vector Functions ............................................................................................... 4767.2. Derivative of a Vector Function with respect to a Scalar ................................ 4767.3. General Rules for Differentiation ..................................................................... 4777.4. Derivative of a Constant Vector ....................................................................... 4797.5. Derivative of a Vector Function in terms of its Components .......................... 479

7.6. If F ( )t has a Constant Magnitude, then F .Fd

dt = 0 ....................................... 480

7.7. If F ( )t has a Constant Direction, then FFd

dt = 0 ....................................... 480

7.8. Geometrical Interpretation of d rdt

................................................................... 480

7.9. Velocity and Acceleration.................................................................................. 4817.10. Scalar and Vector Fields ................................................................................... 4877.11. Gradient of a Scalar Field ................................................................................. 4877.12. Geometrical Interpretation of Gradient ........................................................... 4877.13. Directional Derivative ....................................................................................... 4887.14. Properties of Gradient ....................................................................................... 4887.15. Divergence of a Vector Point Function ............................................................. 4937.16. Curl of a Vector Point Function ........................................................................ 4937.17. Physical Interpretation of Divergence.............................................................. 4947.18. Physical Interpretation of Curl ........................................................................ 4957.19. Properties of Divergence and Curl ................................................................... 4967.20. Repeated Operations by ................................................................................ 4987.21. Integration of Vector Functions ........................................................................ 5047.22. Line Integrals .................................................................................................... 5067.23. Circulation ......................................................................................................... 5077.24. Work Done by a Force ....................................................................................... 5077.25. Surface Integrals ............................................................................................... 5107.26. Volume Integrals ............................................................................................... 5117.27. Gauss Divergence Theorem

(Relation between Surface and Volume Integrals) ........................................... 5167.28. Green’s Theorem in the Plane .......................................................................... 5237.29. Stoke’s Theorem (Relation between Line and Surface Integrals) .................... 526

8. Curvilinear Co-ordinates .......................................................................... 533–5488.1. Definitions ......................................................................................................... 5338.2. Unit Vectors in Curvilinear System ................................................................. 5338.3. Arc Length and Volume Element ..................................................................... 535

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8.4. Gradient in Orthogonal Curvilinear Co-ordinates .......................................... 5388.5. Divergence in Orthogonal Curvilinear Co-ordinates ....................................... 5388.6. Curl in Orthogonal Curvilinear Co-ordinates .................................................. 5398.7. Laplacian in Terms Of Orthogonal Curvilinear Co-ordinates ........................ 5408.8. Special Curvilinear Co-ordinate Systems ........................................................ 5408.9. Some More Special Curvilinear Co-ordinate Systems ..................................... 547

9. Infinite Series ............................................................................................. 549–5979.1. Sequence ............................................................................................................ 5499.2. Real Sequence .................................................................................................... 5499.3. Range of a Sequence .......................................................................................... 5499.4. Constant Sequence ............................................................................................ 5499.5. Bounded and Unbounded Sequences ............................................................... 5499.6. Convergent, Divergent and Oscillating Sequences .......................................... 5509.7. Monotonic Sequences ........................................................................................ 5519.8. Limit of a Sequence ........................................................................................... 5519.9. Every Convergent Sequence is Bounded .......................................................... 5519.10. Convergence of Monotonic Sequences .............................................................. 5529.11. Infinite Series .................................................................................................... 5549.12. Series of Positive Terms .................................................................................... 5549.13. Alternating Series ............................................................................................. 5549.14. Partial Sums ...................................................................................................... 5549.15. Behaviour of an Infinite Series ......................................................................... 5559.16. Absolute Convergence of a Series ..................................................................... 5919.17. Every Absolutely Convergent Series is Convergent ........................................ 5939.18. Uniform Convergence of Series of Functions ................................................... 595

10. Fourier Series ............................................................................................ 598–64810.1. Periodic Functions ............................................................................................. 59810.2. Fourier Series .................................................................................................... 59810.3. Euler’s Formulae ............................................................................................... 60110.4. Dirichlet’s Conditions ........................................................................................ 60210.5. Fourier Series for Discontinuous Functions .................................................... 61410.6. Change of Interval ............................................................................................. 61910.7. Half Range Series .............................................................................................. 62510.8. Fourier Series of Different Waveforms ............................................................ 63710.9. Parseval’s Identity ............................................................................................ 63910.10. Root Mean Square Value (r.m.s. Value) ........................................................... 64010.11. Complex Form of Fourier Series ....................................................................... 64210.12. Practical Harmonic Analysis ............................................................................ 644

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11. Differential Equations of First Order ....................................................... 649–68211.1. Definitions (Differential Equations) ................................................................. 64911.2. Geometrical Meaning of a Differential Equation of the First Order

and First Degree ................................................................................................ 65011.3. Formation of a Differential Equation ............................................................... 65111.4. Solution of Differential Equations of the First Order and First Degree ........ 65411.5. Variables Separable Form ................................................................................ 65411.6. Homogeneous Equations ................................................................................... 65611.7. Equations Reducible to Homogeneous Form ................................................... 65911.8. Linear Differential Equations .......................................................................... 66111.9. Equations Reducible to the Linear Form (Bernoulli’s Equation) .................... 66411.10. Exact Differential Equations ............................................................................ 66711.11. Theorem ............................................................................................................. 66711.12. Equations Reducible to Exact Equations ......................................................... 67011.13. Differential Equations of the First Order and Higher Degree ........................ 67511.14. Equations Solvable for p ................................................................................... 67511.15. Equations Solvable for y ................................................................................... 67811.16. Equations Solvable for x ................................................................................... 67911.17. Clairaut’s Equation ........................................................................................... 681

12. Applications of Differential Equations of First Order ............................ 683–70712.1. Introduction ....................................................................................................... 68312.2. Geometrical Applications .................................................................................. 68312.3. Orthogonal Trajectories .................................................................................... 68712.4. Working Rule to Find the Equation of Orthogonal Trajectories ..................... 68812.5. Physical Applications ........................................................................................ 69112.6. Application to Electric Circuits ........................................................................ 69912.7. Conduction of Heat ............................................................................................ 70012.8. Rate of Growth or Decay ................................................................................... 70112.9. Newton’s Law of Cooling ................................................................................... 70212.10. Chemical Reactions and Solutions ................................................................... 703

13. Linear Differential Equations ................................................................... 708–74413.1. Definitions (Linear Differential Equations) ..................................................... 70813.2. The Operator D.................................................................................................. 70813.3. Theorems ........................................................................................................... 70913.4. Auxiliary Equation (A.E.) ................................................................................. 70913.5. Rules for Finding the Complementary Function ............................................. 710

13.6. The Inverse Operator 1

f ( )D .............................................................................. 713

13.7. Rules for Finding the Particular Integral ........................................................ 71413.8. Method of Variation of Parameters to Find P.I . ............................................. 727

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13.9. Cauchy’s Homogeneous Linear Equation ........................................................ 72813.10. Legendre’s Linear Equation ............................................................................. 73013.11. Simultaneous Linear Equations with Constant Co-efficients ......................... 73313.12. Total Differential Equations ............................................................................. 73813.13. Method for Solving Pdx + Qdy + Rdz = 0 ......................................................... 738

13.14. Solution of Simultaneous Equations of the Form dx dy dzP Q R

.................... 741

14. Applications of Linear Differential Equations ........................................ 745–77814.1. Introduction ....................................................................................................... 74514.2. Simple Harmonic Motion (S.H.M.) ................................................................... 74514.3. Mechanical and Electrical Oscillatory Circuits ............................................... 74814.4. Simple Pendulum .............................................................................................. 76514.5. Gain or Loss of Beats ........................................................................................ 76614.6. Deflection of Beams ........................................................................................... 76914.7. Boundary Conditions ........................................................................................ 77014.8. Applications of Simultaneous Linear Differential Equations ......................... 774

15. Special Functions and Series Solution of Differential Equations ........ 779–86115.1. Gamma Function ............................................................................................... 77915.2. Reduction Formula for (n) .............................................................................. 779

15.3. Value of ( )12 ...................................................................................................... 780

15.4. Beta Function .................................................................................................... 78115.5. Symmetry of Beta Function i.e., B(m, n) = B(n, m) ......................................... 78115.6. Relation between Beta and Gamma Functions ............................................... 781

15.7. To Evaluate 0

2/sin . cosz p qx x dx ; p > – 1; q > – 1 .......................................... 782

15.8. Elliptic Integrals ............................................................................................... 78915.9. Applications of Elliptic Integrals ...................................................................... 79015.10. Error Function ................................................................................................... 79215.11. Series Solution of Differential Equations ........................................................ 79515.12. Definitions ......................................................................................................... 79515.13. Power Series Solution, When x = 0 is an Ordinary Point of the Equation

d y

dx

2

2 + P(xdydx

) + Q(x) y = 0 ............................................................................. 796

15.14. Frobenius Method : Series Solution When x = 0 is a Regular Singular Point of theDifferential Equation ........................................................................................ 803

15.15. Legendre’s Differential Equation ..................................................................... 81815.16. Legendre’s Function of First kind Pn(x) ........................................................... 81915.17. Legendre’s Function of Second kind Qn(x) ....................................................... 820

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15.18. Solution of Legendre’s equation........................................................................ 82015.19. Generating Function for Pn(x) ........................................................................... 82015.20. Rodrigue’s Formula ........................................................................................... 82315.21. Recurrence Relations ........................................................................................ 82615.22. Beltrami’s Result ............................................................................................... 82815.23. Orthogonality of Legendre Polynomials ........................................................... 82815.24. Laplace’s Integral of First Kind ........................................................................ 83015.25. Laplace’s Integral of Second Kind .................................................................... 83015.26. Cristoffel’s Expansion Formula ........................................................................ 83115.27. Cristoffel’s Summation Formula ...................................................................... 83215.28. Expansion of a Function in a Series of Legendre Polynomials

(Fourier-Legendre Series) ................................................................................. 83215.29. Bessel’s Differential Equation .......................................................................... 83815.30. Solution of Bessel’s Equation ............................................................................ 83815.31. Series Representation of Bessel functions ....................................................... 84215.32. Recurrence Relations for Jn(x) .......................................................................... 84315.33. Generating Function for Jn(x) ........................................................................... 85015.34. Integral Form of Bessel Function ..................................................................... 85115.35. Equations Reducible to Bessel’s Equation ....................................................... 85415.36. Modified Bessel’s Equation ............................................................................... 85615.37. Ber and Bei Functions ...................................................................................... 85715.38. Orthogonality of Bessel Functions ................................................................... 85815.39. Fourier-bessel Expansion of F(x) ...................................................................... 859

16. Partial Differential Equations ................................................................... 862–90016.1. Introduction ....................................................................................................... 86216.2. Formation of Partial Differential Equations ................................................... 86216.3. Definitions ......................................................................................................... 86616.4. Equations Solvable by Direct Integration ........................................................ 86816.5. Linear Partial Differential Equations of the First Order ............................... 87016.6. Lagrange’s Linear Equation ............................................................................. 87016.7. Working Method ................................................................................................ 87116.8. Non-linear Equations of the First Order .......................................................... 87616.9. (a) Equations of the Form f(p, q) = 0 ................................................................ 87616.9. (b) Equations of the Form z = px + qy + f(p, q) ................................................. 87816.9. (c) Equations of the Form f (z, p, q) = 0 ............................................................ 87816.9. (d) Equations of the Form f1 (x, p) = f2 (y, q) ..................................................... 88016.10. Charpit’s Method ............................................................................................... 88216.11. Homogeneous Linear Equations with Constant Co-efficients ........................ 88416.12. Rules for Finding the C.F. ............................................................................... 88516.13. Rules for Finding the P.I. .................................................................................. 887

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A Textbook of Engineering Mathematicsby NP Bali and Dr Manish Goyal

Publisher : Laxmi Publications ISBN : 9788131808320 Author : NP Bali and DrManish Goyal

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