Solving first order ODEs system (not autonomous)

Question

The task is to solve this system of ODEs: $$ \begin{cases} & x'(t)=x\sin t\\ & y'(t)=xe^{\cos t} \end{cases} $$

My problem is I have only been shown how to proceed when the system is autonomous. Here however both $x \sin t$ and $x e^{\cos t}$ have $t$ associated with them.

I've been looking for some examples on SE but most of them consist method using eigenvalues - I haven't been introduced to that yet so this case must be much simpler.

What can I do?

Answer 1

$$\begin{cases} & x'(t)=x\sin t\\ & y'(t)=xe^{\cos t} \end{cases}$$

Solve first equation $$(\ln x) '=\sin t \implies \ln x =-\cos t +c$$ $$x(t)=c_1e^{-\cos t}$$ Plug $x$ in the second DE. Then solve the DE: $$ y'(t)=xe^{\cos t}$$ $$ y'(t)=c_1e^{-\cos t}e^{\cos t}=c_1$$ $$y(t)=c_1t+c_2$$

Answer 2

The first equation is quite simple to integrate $$ x'(t)=x(t)\,\sin (t) \implies x(t)=c_1 e^{-\cos (t)}$$ $$y'(t)=x(t)\,e^{\cos(t)} \implies y'(t)=???$$