Answer
True
Work Step by Step
The statement is exactly the same with Theorem 8.7.6, Variation of Parameters.
Consider
$y^{(n)} + a_1(x)y^{(n−1)} +···+ a^{n-1} y′ + a_n(x)y = F(x)$
where $a_1, a_2,..., a_n, F$ are assumed to be (at least) continuous on the interval $I$.
Let $\{y1, y2,..., yn\}$ be a linearly independent set of solutions to the associated homogeneous equation $y^{(n)} + a_1(x)y^{(n−1)} +···+ a^{n-1} y′ + a_n(x)y = F(x)$ on $I$. Then a particular solution is
$y_p = u_1 y_1 + u_2 y_2 +···+ u_n y_n$
