Toronto Math Forum

here is my solution.
$\begin{bmatrix} x'\\ y' \end{bmatrix}=\begin{bmatrix} 1 &-2 \\ 1&-1 \end{bmatrix}+\begin{bmatrix} sec(t)\\0 \end{bmatrix}$
so $det\begin{bmatrix} 1-\lambda &-1 \\ 1& -1-\lambda \end{bmatrix}=0$
$\lambda=i/-i$ and eigenvector for i is $\begin{pmatrix} 1+i\\ 1 \end{pmatrix}$
so $\begin{bmatrix} x'\\ y' \end{bmatrix}=C1e^{it}\begin{bmatrix} 1+i\\ 1 \end{bmatrix}$
$=C1(cost+isint)\binom{1+i}{1}$
$=C1\begin{bmatrix} cost-sint\\ cost \end{bmatrix}+C2i\begin{bmatrix} cost+sint\\ sint \end{bmatrix}$
For non homo:
$\varphi=\begin{bmatrix} cost-sint &cost+sint \\ cost& sint \end{bmatrix}$
Since $\varphi u'=g$
So $\begin{bmatrix} cost-sint &cost+sint \\ cost&sint \end{bmatrix}\begin{bmatrix} u1'\\ u2' \end{bmatrix}=\begin{bmatrix} 1/cost\\ 0 \end{bmatrix}$
so $u1'=-\frac{sint}{cost}$  which means u1=ln(cost)+C2
u2'=1 wihich means u2=t+C1
so the solution is $\begin{bmatrix} x'\\y' \end{bmatrix}=(t+C1)\binom{cost-sint}{cost}+(ln(cost)+C2)\binom{cost+sint}{sint}$