= Solution
Write the system as $\mathbf x'=M\mathbf x$, with $M=\begin{pmatrix}2&5\\-1&-2\end{pmatrix}$. Its <characteristic polynomial> is $\lambda^2+1$, so its <eigenvalues> are $i$ and $-i$. An <eigenvector> for $i$ is $v=(5,-2+i)^{\mathsf T}$. Thus $e^{it}v$ is a complex solution, and its <real part> and <imaginary part> give real solutions:
$$
u(t)=\begin{pmatrix}5\cos t\\-2\cos t-\sin t\end{pmatrix},\qquad w(t)=\begin{pmatrix}5\sin t\\\cos t-2\sin t\end{pmatrix}.
$$
Their initial vectors are $(5,-2)^{\mathsf T}$ and $(0,1)^{\mathsf T}$, with determinant $5\ne0$. Consequently they are <linearly independent> and span the two-dimensional solution space.
\b[The general real solution is]
$$
\boxed{\begin{pmatrix}x(t)\\y(t)\end{pmatrix}=\alpha\begin{pmatrix}5\cos t\\-2\cos t-\sin t\end{pmatrix}+\beta\begin{pmatrix}5\sin t\\\cos t-2\sin t\end{pmatrix}.}
$$
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