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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 ${y_1, y_2}$.