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To find the solution to a non-homogenous, find the complementary function {y_c} to the homogenous solution. Then find a particular solution to find a general solution.

The method of variation of parameters can, in principle, always be used to find a particular solution for a non-homogenous equation. 

Theorem

Given the nonhomogeneous equation {y''+p(x)y'+q(x)y = f(x)}, that has the complementary solution {c_1y_1+c_2y_2} the particular solution can be constructed by

{y_p = -y_1 \int \frac{y_2f(x)}{W(x)}dx + y2 \int \frac{y_1f(x)}{W(x)}dx}

where {W(x)} is the Wronskian of {y_1, y_2}.

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