Find the GS of the System of DE's $\begin{cases} x' = x-3y\\ y'=3x+7y \end{cases}$

Your method is correct.

Another way is to write the system as $X'=MY$

Where $M$ is the matrix $$\begin{pmatrix} 1 & -3 \\ 3 & 7 \end{pmatrix}$$ its eigen values are $m_1,m_2=4$ (repeated roots), so the solution for $x$ can be written as $x=(A t+B)e^{4t}$, putting it in first equation one can get $y$. So there will be two undetermined constants: $A,B$.

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Ordinary Differential Equations

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