I'm trying to find the general solution for the ODE $f''(t)+f(t)=\frac16\sin^3t,\tag{1}$ but something keeps going wrong. I first set $x_1=f$, $x_2=x_1'$, so letting $A=\left[\begin{array}{cc}0 & 1\\-1 & 0\end{array}\right],\quad x=\left[\begin{array}{c}x_1\\x_2\end{array}\right],\quad\text{and}\quad b=\left[\begin{array}{c}0\\\frac16\sin^3t\end{array}\right],$ we have that $(1)$ is equivalent to the nonhomogeneous linear system $x'=Ax+b.\tag{2}$ Observing that $[i,-1]^T$ and $[i,1]^T$ are eigenvectors of $A$ corresponding (respectively) to the eigenvalues $i,-i$, I concluded (and confirmed) that the general solution to the system $x'=Ax$ has the form $d_1e^{it}\left[\begin{array}{c}i\\-1\end{array}\right]+d_2e^{-it}\left[\begin{array}{c}i\\1\end{array}\right]$ for some constants $d_1,d_2$. Equivalently, if $\Phi=\left[\begin{array}{cc}ie^{it} & ie^{-it}\\-e^{it} & e^{-it}\end{array}\right]\quad\text{and}\quad d=\left[\begin{array}{c}d_1\\d_2\end{array}\right]$ for some constants $d_1,d_2$, then $x=\Phi d$ is a solution to $x'=Ax$.
Now, suppose that $\hat c=[\hat c_1,\hat c_2]^T$, where $\hat c_1=\hat c_1(t),\hat c_2=\hat c_2(t)$ are differentiable functions with constant term $0$, and let $\hat x=\Phi\hat c+\Phi d$ for some constant vector $d$. Noting that $A\Phi=\left[\begin{array}{cc}-e^{it} & e^{-it}\\-ie^{it} & -ie^{-it}\end{array}\right]=\Phi',$ it follows that $(\Phi\hat c)'=\Phi'\hat c+\Phi\hat c'=A\Phi\hat c+\Phi\hat c',$ so since $A\Phi d=(\Phi d)'$ by the work done with the homogeneous system, then $\hat x'=(\Phi\hat c)'+(\Phi d)'=A\Phi\hat c+\Phi\hat c'+A\Phi d=A\hat x+\Phi\hat c'.$ Thus, $\hat x'=A\hat x+b\quad\text{if and only if}\quad\Phi\hat c'=b.$
Now, $\Phi$ is invertible, so letting $c$ be the antiderivative (with respect to $t$) of $\Phi^{-1}b$ without integration constant, we should have that $x=\Phi c+\Phi d$ is the general solution to $(2)$, and then $x_1$ would be the general solution to $(1)$, yes? If that's all good, then I'm apparently just making a calculation error when taking the antiderivative or doing the subsequent matrix operations....
Edit: Does anybody have a different, arguably better approach to take for this problem?