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**y'' + y = f(x) -- Variation of Parameters?**

## Homework Statement

Use variation of parameters to solve

##y'' + y = f(x), y(0) = y'(0) = 0.##

## Homework Equations

A description of the method is here: http://en.wikipedia.org/wiki/Variation_of_parameters

## The Attempt at a Solution

The complementary function is:

##y_{CF} = A\cos(x) + B\sin(x) = Ay_1 + By_2##, say.

If ##y = A(x)y_{1}(x) + B(x)y_{2}(x)## is the solution of the forced equation, then, if we choose ##A'y_1 + B'y_2 = 0##, ##y' = Ay'_{1} + By'_{2}## and ##y'' = A'y'_{1} + B'y'_{2}##.

Since ##y_{1,2}## satisfy the homogeneous equation, substitution will lead to ##A'y'_{1} + B'y'_{2} = f(x)##.

Solving these two equations for A' and B' yields

##A'(y_{2}y'_{1} - y_{1}y'_{2}) = f(x)y_{2}## and

##B'(y_{2}y'_{1} - y_{1}y'_{2}) = -f(x)y_{2}##

Now, if ##y_{1} = \cos(x), y_{2} = \sin(x)## then differentiating and substituting gives us

##A' = -f(x) \implies A = \int -f(x) dx## and

##B' = f(x) \implies B = \int f(x) dx##.

But from here, I'm stuck. Any advice?

EDIT: Am I going about this the wrong way? I'm following the example at the top of page 3 here.

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