I have a system of 2 linear diff equations but with a variable coefficients:
$$f''(x)+af'(x)+(1+x)g'(x)+bg(x)=0\\g''(x)+ag'(x)+(1-x)f'(x)-bf(x)=0$$
where $a,b$ are some positive constants. I have no idea how to solve this system (or even if it has an anayltic solution), I'll appreciate any help or some reference.
HINT: Let $f'(x)=h(x)$ and $g'(x)=k(x)$. The system of second order ODE can be reduced to an homogeneous system of first order with variable coefficients $z'(x)+A(x)z(x)=0$: $$ \begin{pmatrix} f'\\ g'\\ h'\\ k' \end{pmatrix}+ \underbrace{ \begin{pmatrix} 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1\\ 0 & 0 & a & 1+x\\ -b & 0 & 1-x & a\\ \end{pmatrix}}_{A(x)}\ \underbrace{ \begin{pmatrix} f\\ g\\ h\\ k \end{pmatrix}}_{z(x)} =\begin{pmatrix} 0\\ 0\\ 0\\ 0 \end{pmatrix} $$ Thus you can solve it by Fundamental System of Solutions and Fundamental Matrix Method (Wronskian and Liouville's Formula) or by Method of Variation of Constants (Lagrange Method)