Find the solution for a boundary value problem

Question

Please, how can we find the solution of this second order boundary value problem $$-(e^{-2x}u')'-\ln(x^2+2)u= 2 e^ {-2x} - x \ln(x^2+2),\,\, x\in ]0,1[, u(0)=0,u(1)=1?$$

Or more generally, What's the change of variable we can use to find the exact solution of the problem $$-(P(x)u')'+q(x)u=f(x),x\in ]0,1[, u(0)=0,u(1)=1$$ ?

Help me please.

Answer 1

$u_1(x)=x$ is a solution. You can find the second (linearly independent) solution by using reduction of order, that is, set $u_2(x)= xv$ and determine $v$.