Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 24
Variation of Parameters
The method of variation of parameters gives an explicit formula for
a particular solution to a linear differential equation once all of the homogeneous
solutions are known. The particular solution is a pseudo-linear
combination of the homogeneous equation. By a pseudo-linear combination
we mean an expression that has the same form as a linear combination,
but the constants are allowed to depend on t:
y P = u 1 (t)y 1 + u 2 (t)y 2 (24.1)
where u 1 (t) and u ( t) are unknown functions of t that are treated as parameters.
The name of the method comes from the fact that the parameters
(the functions u 1 and u 2 in the linear combination) are allowed to vary.
For the method of variation of parameters to work we must already know
two linearly independent solutions to the homogeneous equations
Suppose that y 1 (t) and y 2 (t) are linearly independent solutions of
a(t)y ′′ + b(t)y ′ + c(t)y = 0 (24.2)
The idea is to look for a pair of functions u(t) and v(t) that will make
a solution of
Differentiating equation (24.3)
y P = u(t)y 1 + v(t)y 2 (24.3)
a(t)y ′′ + b(t)y ′ + c(t)y = f(t). (24.4)
y ′ P = u ′ y 1 + uy ′ 1 + v ′ y 2 + vy ′ 2 (24.5)
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