I have a system of differential equations which I do not know how to solve: $$v_1'=\frac{v_1}{\sqrt{v_1^2+v_2^2}}*a$$ $$v_2'=\frac{v_2}{\sqrt{v_1^2+v_2^2}}*a+g$$ It comes from this equation of motion: $\frac{dv}{dt}=\frac{1}{|\vec{v}|}\vec{v}a+\vec{g}$ with $\vec{g}=\left(\begin{array}{c} 0 \\ 9.81 \end{array}\right)$. I wanted the acceleration to follow the direction of the velocity, given an initial velocity and a nuisance factor $g$ to change the direction over time.
I had systems of linear equations in school but never learned anything about vectors or differentiation (I am 10th grade in Germany so hopefully we will do this in future). I googled my knowledge together so I hope you understand what I want to do.
I tried to substitute one into the other but this does not work because there are 4 unknowns $v_1$, $v_1'$, $v_2$, $v_2'$ and I can only eliminate $v_1$ or $v_1'$ not both.
Second I saw the usage of a coefficients-matrix but every time only for linear functions so I decided not to learn and try this method.
I would already be happy if someone could recommend a source to learn about systems of differential equations.