Consider $$ \begin{pmatrix} M_1' \\ M_2' \end{pmatrix} = \begin{pmatrix} g_1 & g_1 \\ f_2 & g_2 \end{pmatrix} \begin{pmatrix} M_1 \\ M_2 \end{pmatrix}, $$ where $f,g, M$ are unknown real-valued function of one variable.
Q1: Is it solvable, if the initial points are non zero?
Q2: I ran sympy to solve the equation, and had the form $$ m_1(t) = C_1 x_0(t) + C_2 x_0(t) \int \frac{f_1(t) e^{\int g_1(t) dt} e^{\int g_2(t) dt}}{x_0^2(t)} dt $$ and $$ m_2(t) = C_1 y_0(t) + C_2 \left( y_0(t) \int \frac{f_1(t) e^{\int g_1(t) dt} e^{\int g_2(t) dt}}{x_0^2(t)} dt + \frac{e^{\int g_1(t) dt} e^{\int g_2(t) dt}}{x_0(t)} \right) $$ What are $x_0$ and $y_0$?