How to solve this system of ODE's?

Question

I'm not sure how to proceed to solve this system of ODE's;

$$ \begin{bmatrix}\dot{x}_1 \\\dot{x}_2\end{bmatrix}=\begin{bmatrix} \cos t & -\sin t\\ \sin t & \cos t \end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}$$

Thank you.

Answer 1

This is a linear system of the form $$ x'=A(t)x, $$ where $x\in\mathbb R^2$, $A(t)\in C(\mathbb R;\mathbb R^{2\times 2})$, and most important $$ A(s)A(t)=A(t)A(s). \tag{1} $$ Satisfaction of $(1)$ implies that the solution of $$ x'=A(t)x,\,\,\,x(0)=\xi_0, $$ is equal to $$ x(t)=\exp \Big(\int_{0}^t A(s)\,ds\Big)\,\xi_0. $$ In our case $$ A(t)=\left(\begin{matrix}\cos t & -\sin t\\ \sin t&\cos t\end{matrix}\right), $$ then $$ \int_0^tA(s)\,ds=\left(\begin{matrix}\sin t & \cos t-1\\ 1-\cos t&\sin t\end{matrix}\right). $$ Next we use use the fact that $$ \exp\left(\begin{matrix}a & -b\\ b&a\end{matrix}\right)= \left(\begin{matrix}\mathrm{e}^a\cos b & -\mathrm{e}^a\sin b\\ \mathrm{e}^a\sin b&\mathrm{e}^a\cos b\end{matrix}\right), $$ and finally we obtain that $$ x(t)=\left(\begin{matrix}\mathrm{e}^{\sin t}\cos(1-\cos t) & -\mathrm{e}^{\sin t}\sin(1-\cos t)\\ \mathrm{e}^{\sin t}\sin(1-\cos t)&\mathrm{e}^{\sin t}\cos(1-\cos t)\end{matrix}\right)\,\xi_0. $$