I have a system of differential equations, and I want to know if it is possible to get an analytical solution before resorting to numerical methods and software.
\begin{align} \frac{dy_1}{dx} + y_1 - y_3 &= 0 \\ \frac{dy_2}{dx} + y_2 - y_4 &= 0 \\ \frac{dy_3}{dx} + y_1 - y_3 &= f(x)\frac{y_3}{y_3 +1}\frac{y_4}{y_4 +1} \\ \frac{dy_4}{dx} + y_2 - y_4 &= f(x)\frac{y_3}{y_3 +1}\frac{y_4}{y_4 +1} \end{align}
where $f(x)$ is a simple, probably linear, function
I think because of the non linear term, a solution by the laplace or eigenvalue methods is ruled out. Is there something I'm missing, or is it really only possible numerically?
Substract equation 2 and 1, then substract 3 and 4.
You find this system $$ \begin{cases} z'-z+v=0 \\ v'+v-z=0 \end{cases} $$
Where $z=y_3-y_4$ and $v=y_1-y_2$ $$ \begin{cases} v''=0 \implies v=k_1x+k_2-k_1 \implies y_1=y_2+k_1x+k_2-k_1 \\ z'' =0\implies z=k_1x+k_2 \implies y_3=y_4+k_1x+k_2 \end{cases} $$ With theses results you can solve your system ( take the first two equations)..... $$ \begin{cases} dy_1/dx + y_1 - y_3 = 0 \\ dy_2/dx + y_2 - y_4 = 0 \end{cases} $$