I have the following : $$ Y'(t)=JY(t)+K(t) $$ where $Y, J, K$ are matrices. The text book I have tells me the following for $J=J_3(\lambda)$: $$\begin{align} y'_1(t)&=\lambda y_1(t)+K_1(t)\\ y'_2(t)&=\lambda y_2(t)+(y_1(t)+K_2(t))\\ y'_3(t)&=\lambda y_3(t)+(y_2(t)+K_3(t)) \end{align}$$
If I have $$K(t)= \begin{pmatrix} 3\\ e^t-6\\ e^{-3t}-e^t+3 \end{pmatrix}$$
then am I correct in setting up the first equation as follows: If $\lambda=-1$, $y'_1(t)=-y_1(t)+3$, then I could use the integrating factor of $e^t$ and solve to get: $y_1$(t)=3+c$e^{-t}$
Is this correct so far?
Can someone please step me through getting $y_2(t)$?