My Problem is this given system of differential equations: $$y_{1}^{\prime}=y_{1}-y_{2}$$ $$y_{2}^{\prime}=5y_{1}+3y_{2}$$ I am looking for the solution.
My Approach was: this seems to be a system of first-order differential equations. they are ordinary differential equations.
I built the corresponding matrix:
$$\underbrace{\pmatrix{ y_1^{\prime} \\ y_2^{\prime}}}_{\large{ {\vec y^{\prime}}}} = \underbrace{\pmatrix{1 & -1 \\ 5 & 3}}_{\large{\mathbf A}}\underbrace{\pmatrix{y_1\\y_2}}_{\large{\vec y}}$$ Now i need to find the eigenvalues of this matrix in order to determine the eigenvectors And here i am stuck. I failed in finding the eigenvalues. Every eigenvalue i find seems to be no number. so cannot calculate with it. But if the eigenvalues are anything other than numbers, (for example a complex number) how can i find the solution for the system of differential equations in this case?