Finding a basis of solutions to a linear homogeneous system with a matrix with non-constant entries. Is there a way to find a basis of solutions to the following linear homogeneous system using the eigenvector method?
\begin{array}{l} \\ {\qquad \boldsymbol{y}^{\prime}(t)=A(t) \boldsymbol{y}(t), \quad A(t)=\left[\begin{array}{cc}{2 t /\left(t^{2}+1\right)} & {0} \\ {2 t} & {2 t}\end{array}\right]}\end{array}
I've tried to but it gets quite unusual, do the matrix entries have to be constants for this method? I'm going to guess that the matrix exponential won't work either because of this.
I've found the solution by just doing the individual vector components, but was just wondering if there was a method to do it by leaving it in "matrix form".

No, it's not a matrix exponential, though it's sometimes called a "time-ordered exponential", and eigenvectors/eigenvalues won't help.  In general there's no way to do it in closed form, though as you mentioned, in this case you can solve it by doing $y_1$ first and then $y_2$.
