Page 120 - Schaum's Outline of Differential Equations
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CHAPTER 12
Variation of
Parameters
Variation ol' parameters is another method (see Chapter 1 1) for finding a particular solution of the nth-order
linear differential equation
once the solution of the associated homogeneous equation L(v) = 0 is known. Recall from Theorem 8.2 thai if
y\(x), VjOt). .... y,,(.v) are n linearK independent solutions of L(v) = 0. then Ihe general solution of L(y> = 0 is
THE METHOD
A particular solution of L(y) = </>ix) has the form
where \'j = yj(x) (' = '-2, ••• • "I is gi\en in Kq. (12.2} and i\- (i = I. 2, ... . n) is an unknown function of .t \\hidi
Slill must he determined.
1
To find v/, firsl solve the following linear equations simultaneous!} for i ,';
1
Then integrate each r,' to obtain ,. disregarding all constants of integration. This is permissible because we are
i
seeking only one particular solution.
Example 12.1. For the special case n = .l. Eqs. (12.4) reduce to
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