# 4.6 | Variation of Parameters

Variation of Parameters If $$y_1$$ and $$y_2$$ are two linearly independent solutions to the homogeneous equation
$$y^{\prime \prime}+p(t) y'(t) + q(t) y(t) = 0$$
on an interval $$I$$ where $$p(t), q(t),$$ and $$g(t)$$ are continuous, then a particular solution to the nonhomogeneous equation
$$y^{\prime \prime}+p(t) y'(t) + q(t) y(t) = g(t)$$
is given by $$y_p = v_1 y_1 + v_2 y_2,$$ where $$v_1$$ and $$v_2$$ are determined up to a constant by the pair of equations
\begin{aligned} y_1 v_1^{\prime} + y_2 v_2^{\prime} &= 0, \\ y_1^{\prime} v_1^{\prime} + y_2 ^{\prime} v_2^{\prime} & = g, \end{aligned}
which have the solution
$$v_1(t) = \int \frac{-g(t) y_2(t)}{W[y_1,y_2](t)} \; dt$$
and
$$v_2(t) = \int \frac{g(t) y_1(t)}{W[y_1,y_2](t)} \; dt.$$