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
$$
v_2(t) = \int \frac{g(t) y_1(t)}{W[y_1,y_2](t)} \; dt.
$$

E 4.6 Exercises