7/28/2019 Spring 2012_MTH401_2

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Assignment # 02

MTH401 (Spring 2012)

 

Total marks: 30Lecture # 12-18

Due date: 08-05-2012

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7/28/2019 Spring 2012_MTH401_2

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Question#1 Marks 10

In the following differential equation the indicated function ( )1 y x is a solution of the

associated homogeneous equation. Use the method of reduction of order to find a second

solution ( )2 y x of the homogeneous equation and a particular solution of the given

nonhomogeneous equation using method of undetermined coefficients-superpositionapproach.

  1

22

24 2 ;

xd y y y e

dx

−− = =

Question#2 Marks 20

Solve the following initial value problem.

  ( ) ( )0 ; 0 1dy

 p x y ydx

+ = =

Where

  ( )2 0 1

1 1

 x p x

 x

≤ ≤

>=

Hint:

Linear differential equations sometimes occur in which the function ( ) p x have

 jump discontinuities. If  0 x is such a point of discontinuity, then it is necessary to solve

the equation separately for  0 x x< and 0 x x> . Afterwards, the two solutions are

matched so that the function ( ) y x is continuous at 0 x ; this is accomplished by a proper 

choice of the arbitrary constants.

In the given differential equation ( ) p x has a jump discontinuity at 01 x = , then it is

necessary for all of you to solve the equation separately for  1 x < and 1 x > .