04/18/23Method of Variation of Parameters Chapter 4 1

04/18/23 2Method of Variation of Parameters Chapter 4

The general solution of the non - homogeneous differential equation

There are two parts of the solution:

1. solution of the homogeneous part of DE

2. particular solution

( )ay by cy f x

cy

py

c py y y General Solution

0ay by cy

04/18/23Method of Variation of Parameters Chapter 4 3

General solution

c py y y

Complementary Function, solution of Homgeneous part

Particular Solution

0ay by cy Due to f(x)

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This method can be applied to any non-homogeneous differential equation no matterWhat the coefficients and the function f(x) are.

Method of undetermined coefficients is mostly quicker and easier to apply as it involves differentiation. Whereas method of variation of parameters involve Integration. We may prefer to use method of undetermined coefficients where ever possible.

( )ay by cy f x

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To apply method of variation of parameters

STEP: 1.

Step:2. Identify

Step:3.

1 1 2 2cy c y c y

1 2

1 2

1 2

( ), , , ( )

( )

y y g xy y W f x

a xy y

21

12

( )

( )

y f xu

wy f x

uw

( )ay by cy g x

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04/18/23Method of Variation of Parameters Chapter 4 9

c

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