Term Paper of Mth102,Sec No-d6905,Roll No-A05 - Copy

7/28/2019 Term Paper of Mth102,Sec No-d6905,Roll No-A05 - Copy

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MTH102

EXACT DIFFERENTIAL

EQUATION

SUBMITTED TO : SUBMITTED

BY :

Miss Ravinder kaur Jagbeer yadavDept. of Mathematics Reg. No:-10906497

ROLL NO:-A05


2/13

SEC :-D6905

ACKNOWLEDGMENT

I would express my gratitude to all those who gave me

the possibility to complete this term work. I want to thankthe department of MTH102 for giving me permission to

commence this work in the first instance and to use the

research data.

I am deeply indebted to my teacher

Miss Ravinder kaur whose suggestions and

encouragement helped me in all time of research for

writing this term work.


3/13

Jagbeer yadav

CONTANT:

EXACT DIFFERENTIAL EQUATIONS

EQUATION REDUCIBLE TO EXACT

EQUATION:

I.F FOUND BY INSPECTION

INTEGRATING FACTORS

SEPARABLE EQUATIONS(I.F OF

HOMOGENEOUS)

PROBLEM

BILIOGRPHY:


4/13

EXACT DIFFERENTIAL EQUATIONS

A differential equation obtained from its primitive directly by

differentiation without any operation of multiplication, elimination orreduction is said to be an exact differential equation.

Thus a differential equation of the form M(X,Y)dx+N(X,Y)dy=0is an

exact differential equation if it can be obtained directly by

differentiating the equation u(x,y)=c which is primitive.

Du=Mdx+Ndy

Mdx+Ndy=0is integ.Mdx+integ(terms of N not containing X)

Theorem: Solutions to Exact Differential Equations

Let M, N, My, and Nx be continuous with

My = Nx

Then there is a function f with

fx = M and fy = N

such that

f(x,y) = C


5/13

is a solution to the differential equation

M(x,y) + N(x,y)y' = 0

A first-order differential equation is one containing a firstbut nohigherderivative of the unknown function. For virtually every such

equation encountered in practice, the general solution will contain

one arbitrary constant, that is, one parameter, so a first-order IVP will

contain one initial condition. There is no general method that solves

every first-order equation, but there are methods to solve particular

typesGiven a functionf(x, y) of two variables, its total differentialdf

is defined by the equation

The DE's that come up in Calculus are Separable. As we just saw this

means they can be

and

.

This means that

so that .

Such a du is called an "Exact", "Perfect" or "Total" differential.


6/13

As we will see in Orthogonal Trajectories (1.8), the expression

represents

a one-parameter family of curves in the plane. For example,

is a family of circles of radius and is a family of

parabolas.

Let us find the differential du for

.

Calculate

(solution)

and ended with

1st order D. E. of the form

This D. E. is called exact if there is some function u(x,y) so that

and, of course, . since

.


7/13

Since and , and If

we make the mild assumption that , are continuous then we get as

a freebie that

Now we can carry out two "partial" integrations:

so

so

Notice that the integration so-called constants each depend on one of the

variables.

Now we do some "criss-crossing" to get our solution .

First, get by solving for dk/dy in

and then carry out a ("partial') integration:

.

EQUATION REDUCIBLE TO EXACTEQUATION:

Differential equation which are not exact can be made exact after

multiplying by a suitable factor called the integrating factor.


8/13

Ydx-xdy=0 ,M=y ,N=-x the equation is not exact.

Multiplying the equation 1/y2 it ydx-xdy/y2 =0 Multiplying the equation 1/x2 it d(y/x)=0

Multiplying the equation 1/xy it dx/x-dy/y=0

Which is exact -1/y2,1/x2,1/xy are integrating factors.

I.F found by inspection :in a number of problem a little

analyisis helps to find the ingrting factor .

1.Ydx+xdy=d(xy)y

dx)y,x(M =

x

dy)y,x(N

Set )y,x(Mx

)y,x(F=

. Integrate with respect to x to get F ),( yx

F ),( yx = dx)y,x(M + )y(

Differentiate with respect to y to get )y,x(Ny

[ dx)y,x(M + )y( ]=

)y,x(N )y( F ),( yx = g(x,y) + )(y , if no boundaryvalue is given.

Integrating Factors

What ifx

)y,x(N

y

)y,x(M

Definition

If 0dy)y,x(Ndx)y,x(M =+ is not exact, but0)y,x(N)y,x(dx)y,x(M)y,x( =+ is exact, then )y,x( is called an

integrating factor.


9/13

Example : Show that ( ) 0dyxdxxy2y 22 =+ is not exact, then find nsuch that yn is an integrating factor.

i ( ) x2y2xy2yy

2 +=+

x2)x(

x

2 =

therefore the DE is

not exact.

ii. Multiply the DE by yn, then solve. 0dyxydx)xy2y(y 2n2n =+

( ) ( ) ( ) n1n1n2n xy1n2y2nxy2yy

+++=+

++must equal ( ) xy2xy

x

n2n =

which

means ( ) 1ny2n ++ must equal 0 and ( ) nxy1n2 + must equal nxy2

for this to be so n must equal -2.

Separable Equations(I.F OF HOMOGENEOUS)

0dy)x(g)x(fdx)y(G)x(F =+ This type of DE is called separable because it

can be written in the form (variables can be separable)

0dy)y(Ndx)x(M =+

The first equation is usually not exact but multiplying it by the

appropriate integrating factor will make it exact, but use of an

integrating factor may eliminate solutions or may lead to extraneous

solutions.After multiplying by the integrating factor )y(G)x(f1

the equation becomes: 0dy)y(G

)y(gdx

)x(f

)x(F=+ where

)x(f

)x(F)x(M =

and)y(G

)y(g)x(N = . Solutions are of the form cdy)y(Ndx)x(M ++

where 0)y(G0)x(f & .


10/13

Problem ( ) ( )3 2 2 24 6 sin sin cos cos 12 6 0xy x x y dx x y y x y dy + + + = ,

( ) 0y =

Step 0: Put the equation into standard form.

( ) ( )3 2 2 24 6 sin sin cos cos 12 6 0xy x x y x y y x y y + + + =

Then ( )3

, 4 6 sin sinM x y xy x x y= and ( )2 2 2

, cos cos 12 6N x y x y y x y= + + .

Step 1: Compute ( ),yM x y and ( ),xN x y .

( ) 2, 12 sin cosyM x y xy x y=

( ) 2, sin cos 12xN x y x y xy= +

Thus ( ) ( ), ,y xM x y N x y .

Step 2: Make a choice between two methods of solving the equation.

Choose one of the two methods. Below each method is presented,

but you need only one of them. The purpose of presenting both methods

is simply to demonstrate that either one will provide the answer.

[First method: ( ) ( ), ,x y M x y dx = ]

Step 3: Perform the integration according to the method chosen in Step

2.

Since the equation is exact, there is a function for which

( ) ( ) 3, , 4 6 sin sinx x y M x y xy x x y = =


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and

( ) ( ) 2 2 2, , cos cos 12 6y x y N x y x y y x y = = + + .

Then ( ) ( ) ( )3

, , 4 6 sin sinx y M x y dx xy x x y dx = = ( )

2 3 22 3 cos sinx y x x y g y= + + .

Now it is given that ( )2 2 2

, cos cos 12 6N x y x y y x y= + + ; and from above

( ) ( )( )2 3 2, 2 3 cos siny x y x y x x y g yy

= + +

( )2 26 cos cosx y x y g y= + + .

Since( ) ( ), ,y x y N x y =

,

( )2 2 2 2 26 cos cos cos cos 12 6x y x y g y x y y x y+ + = + +

( ) 212g y y =

( ) 34g y y=

So the one parameter family of functions that define the solutions of the

equations is2 3 2 3

2 3 cos sin 4x y x x y y C + + = .

[Second method: ( ) ( ), ,x y N x y dy = ]

Step 3: Perform the integration according to the method chosen in Step

2.

Since the equation is exact, there is a function for which

( ) ( ) 3, , 4 6 sin sinx x y M x y xy x x y = =

and


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( ) ( ) 2 2 2, , cos cos 12 6y x y N x y x y y x y = = + + .

Then ( ) ( ) ( )2 2 2

, , cos cos 12 6x y N x y dy x y y x y dy = = + +

( )3 2 3cos sin 4 2x y y x y f x= + + + .

Now it is given that ( )3

, 4 6 sin sinM x y xy x x y= ; and from above

( ) ( )( )3 2 3, cos sin 4 2x x y x y y x y f xx

= + + +

( )

3sin sin 4x y xy f x= + +

Since ( ) ( ), ,x x y M x y = ,

( )3 3sin sin 4 4 6 sin sinx y xy f x xy x x y + + =

( ) 6f x x =

( ) 23f x x=

So the one parameter family of functions that define the solutions of the

equations is

3 2 3 2cos sin 4 2 3x y y x y x C+ + =.

Step 4: Evaluate C using the initial condition.

Using the initial condition ( ) 0y = ,

( ) ( )3 32 2 2

2 0 3 cos sin 0 4 0 3C = + + = .

So the solution of the is

2 3 2 3 2

2 3 cos sin 4 3x y x x y y + + =


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REFRANCE:

www.wolframe.com

www.math.fsu.edu/~fusaro/EngMath/Ch1/SEDE.htm www.ltcconline.net/greenl/courses/204/.../exactDiffEQs.html

www.cliffsnotes.com/.../Exact-Equations.topicArticleId-19736,articleId-19710.htm

chapter form n.p.bali/b.s Grewal.
http://www.wolframe.com/http://www.math.fsu.edu/~fusaro/EngMath/Ch1/SEDE.htmhttp://www.ltcconline.net/greenl/courses/204/.../exactDiffEQs.htmlhttp://www.ltcconline.net/greenl/courses/204/.../exactDiffEQs.htmlhttp://www.ltcconline.net/greenl/courses/204/.../exactDiffEQs.htmlhttp://www.cliffsnotes.com/.../Exact-Equations.topicArticleId-19736,articleId-19710.htmhttp://www.cliffsnotes.com/.../Exact-Equations.topicArticleId-19736,articleId-19710.htmhttp://www.cliffsnotes.com/.../Exact-Equations.topicArticleId-19736,articleId-19710.htmhttp://www.cliffsnotes.com/.../Exact-Equations.topicArticleId-19736,articleId-19710.htmhttp://www.cliffsnotes.com/.../Exact-Equations.topicArticleId-19736,articleId-19710.htmhttp://www.cliffsnotes.com/.../Exact-Equations.topicArticleId-19736,articleId-19710.htmhttp://www.wolframe.com/http://www.math.fsu.edu/~fusaro/EngMath/Ch1/SEDE.htmhttp://www.ltcconline.net/greenl/courses/204/.../exactDiffEQs.htmlhttp://www.cliffsnotes.com/.../Exact-Equations.topicArticleId-19736,articleId-19710.htmhttp://www.cliffsnotes.com/.../Exact-Equations.topicArticleId-19736,articleId-19710.htm

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Term Paper of Mth102,Sec No-d6905,Roll No-A05 - Copy