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01/05/15 3Method 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′′ ′+ + =
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01/05/15Method of Variation of Parameters Chapter 4 4
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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01/05/15Method of Variation of Parameters Chapter 4 5
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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01/05/15Method of Variation of Parameters Chapter 4 8
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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01/05/15Method of Variation of Parameters Chapter 4 9Video.edhole.com
01/05/15Method of Variation of Parameters Chapter 4 10
c
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