Solve an ODE system

Question

There is a system $\dot x = x, \dot y = x+2y$. I got this answers $x=C_1\cdot e^t$ and $y=C_1\cdot e^{2t}-x/2-1/4$ but I find it wrong somehow. Will you explain the full solution to that?

Answer 1

$$\dot x = x, \\ \dot y = x+2y$$ I got this: $$ x'=x \implies x=c_1e^t$$ And: $$y' = x+2y$$ $$y' = c_1e^t+2y$$ $$y'-2y=c_1e^t$$ $$(ye^{-2t})'=c_1e^{-t}$$ Integrate both sides: $$ye^{-2t}=-c_1e^{-t}+c_2$$ $$y=-{c_1}e^t+c_2e^{2t}$$ It's hard to tell you what mistakes you made without posting your full attempt.

Answer 2

Your answer for $x$ is right. Then you are left with solving $\dot{y} = C_1 e^t + 2y$. There are various ways to do it (the variation of constants formula, multiplying by an exponential factor...)

Do you know any of those methods?