Let $V$ is vector space that have function in this form
$\alpha e^{2x} \cos x +\beta e^{2x} \sin x$
where $\alpha , \beta \in \mathbb R$, and linear transformation $L:V\to V$ is going like this:
$L(f)=f´+f$
a)Find a matrix of linear operator using a base
${e^{2x} \cos x,e^{2x} \sin x}$
b) using that solution find a solution for differential equation $y´+y=e^{2x}$
I know for a) it is easy since $L(e^{2x} \cos x)=3e^{2x}\cos x-e^{2x}\sin x$, $L(e^{2x}\sin x)=e^{2x}\cos x+e^{2x}\sin x$
so matrix is
L=$\begin{bmatrix} 3& -1\\ 1& 3 \end{bmatrix}$
for b) I know $f(y)=y´+y=e^{2x}\cos x$, since $y\in V$ then $y=\alpha e^{2x}\cos x+\beta e^{2x}\sin x$,
so $f(\alpha e^{2x}\cos x+\beta e^{2x}\sin x)=e^{2x}\cos x$, now $\alpha f(e^{2x}\cos x)+\beta f( e^{2x}\sin x)=e^{2x}\cos x$,
and I get $\cos x(3\alpha+\beta)+\sin x(3\beta-\alpha)=\cos x$,
but know I know that if I find a solution for$ \alpha$ and $\beta$ I get $y_p=\frac{1}{10}e^{2x}\sin x+\frac{3}{10}e^{2x}\cos x$,
but I do not know how to get homogeneous solution, I know that I need to get $y_h=Ce^{-x}$,
but I have no idea how to get solution I know that I use matrix and find eigenvalues, and eigenvectors for system, and I get $\lambda_1=3+i$ and $\lambda2=3-i$, but somehow I need to get a $\lambda=-1$ but how to get that?, can someone help me please, again sorry i need to change since I can not ask more than 50 question