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UNIVERSITY OF PETROLEUM & ENERGY STUDIES

ASSIGNMENT-3

B.TECH-II SEMESTER, 2014-2015

Course: B. Tech (All Branches) Subject: Mathematics-II

Topic: Differential Equations Code: MATH-102

1. Consider the second order ODE:

𝑦′′ + 𝑃𝑦′ + 𝑄𝑦 = 0. ……………………….. (A)

where 𝑃 and 𝑄 are constants. The substitution of 𝑦 = 𝑒𝑚𝑥 in (A) yields the

auxiliary equation:

𝑚2 + 𝑃𝑚 + 𝑄 = 0.

The two roots of the above quadratic equation are given by

𝑚 =−𝑃±√𝑃2−4𝑄

2

For 𝑃2 − 4𝑄 = 0 , we obtain only one solution as 𝑦 = 𝑒𝑚𝑥 with 𝑚 = −𝑃/2 . Prove

that the second linearly independent solution is given by 𝑥𝑒−𝑃𝑥/2 .

2. Consider the boundary value problem:

𝑦′′ + 𝑦 = 1 , 𝑦(0) = 𝑦(2𝜋) = 0

Show that it has more than one solution. How many of them are linearly

independent ?

3. Let 𝑦(𝑥) be the solution of the ODE : 𝑑2𝑦

𝑑𝑥2 + 2𝑑𝑦

𝑑𝑥+ 𝐵𝑦 = 0. Find lim𝑥→∞ 𝑦(𝑥) for the

following choices of constant 𝐵.

(a) 0 < 𝐵 < 1

(b) 𝐵 > 1

4. Consider the second order ODE:

𝑑2𝑦

𝑑𝑥2 + 𝐴𝑑𝑦

𝑑𝑥+ 𝐵𝑦 = 0. (𝐴 > 0, 𝐵 > 0)

Find the condition for which the given ODE always admits two linearly independent

solutions that are products of exponential and trigonometric functions.

5. Let 𝑓(𝑥) and 𝑥𝑓(𝑥) be the particular solutions of the differential equation:

𝑦′′ + 𝑅(𝑥)𝑦′ + 𝑆(𝑥)𝑦 = 0

Then find the solution of the differential equation 𝑦′′ + 𝑅(𝑥)𝑦′ + 𝑆(𝑥)𝑦 = 𝑓(𝑥).

6. Let 𝑦: ℝ → ℝ be a solution of the ODE

𝑑2𝑦

𝑑𝑥2 − 𝑦 = 𝑒−𝑥, 𝑥 ∈ ℝ ; 𝑦(0) =𝑑𝑦

𝑑𝑥(0) = 0

Prove that:

(a) 𝑦 attains its minimum on ℝ .

(b) lim𝑥→∞ 𝑒−𝑥𝑦(𝑥) =1

4 .

7. Suppose 𝑦1(𝑥) be the known solution of the differential equation:

𝑑2𝑦

𝑑𝑥2 + 𝑃(𝑥)𝑑𝑦

𝑑𝑥+ 𝑄(𝑥)𝑦 = 0.

Show that if 𝑦2(𝑥) = 𝑣(𝑥)𝑦1(𝑥) is the another linearly independent solution then

𝑣(𝑥) = ∫𝑒− ∫ 𝑃 𝑑𝑥

𝑦12(𝑥)

.

8. If 𝑦1 and 𝑦2 are two linearly independent solutions of the homogeneous equation:

𝑦′′ + 𝑃(𝑥)𝑦′ + 𝑄(𝑥)𝑦 = 0.

then show that 𝑃(𝑥) = −𝑦1𝑦2

′′−𝑦2𝑦1′′

𝑊(𝑦1,𝑦2) and 𝑄(𝑥) =

𝑦1′𝑦2

′′−𝑦2′𝑦1

′′

𝑊(𝑦1,𝑦2)

where 𝑊(𝑦1, 𝑦2) = | 𝑦1 𝑦2

𝑦1′ 𝑦2

′ | is the Wronskian determinant.

9. Find the general solution of 𝑦′′ − 𝑥𝑓(𝑥)𝑦′ + 𝑓(𝑥)𝑦 = 0 if it is known that the line

bisecting the positive quadrant of 𝑥𝑦 −plane is its solution.

10. Determine all real numbers 𝐿 > 1 so that the boundary value problem:

𝑥2𝑦′′(𝑥) + 𝑦(𝑥) = 0 , 1 < 𝑥 < 𝐿 ; 𝑦(1) = 𝑦(𝐿) = 0

has a non-trivial solution.