System of differential equations, need help on correcting the answer I get.

Question

I am solving this problem: $$ x'=z-y, y'=z, z'=z-x $$. The method I used involves eigenvectors. Eigenvalues that I found are: 1, i and -i, and the solution I get is x=0, y=ce^x, z=ce^x. Everything would look nice, but wolframalpha tells me that the answer should include sin and cos functions.. Could someone give me a hint on where do they come from? Or my answer is good enough? Any help is appreciated!

Answer 1

That the eigenvalues are $1$, $\mathrm i$ and $-\mathrm i$ means that the solutions are linear combinations of the functions $e_1$, $e_{\mathrm i}$ and $e_{-\mathrm i}$, where $e_\lambda:t\mapsto \mathrm e^{\lambda t}$, for every $\lambda$ in $\mathbb C$. It happens that $e_\mathrm i=c+\mathrm is$ and $e_{-\mathrm i}=c-\mathrm is$, where $c:t\mapsto\cos t$ and $s:t\mapsto\sin t$, hence the vector space generated by $\{e_i,e_{-i}\}$ is also generated by $\{c,s\}$. Thus, the solutions are linear combinations of the functions $e_1$, $c$ and $s$.