How can transform non homogeneous boundary conditions?

Question

How can I transform the non homogeneous boundary conditions to homogeneous boundary conditions in the the following ordinary differential equation? $$ \begin{cases} y’’’’’(x)= e^{-x} \cdot y^2(x) \\ y(0)=1, \; y(1)=e, \\ y’(0)=1, \;y’(1)=e, \\ y''(0)=1, \\ \end{cases} $$

Answer 1

Write $y=u+v$, where $v$ is an arbitrary function satisfying the given boundary conditions. One possibility is $v(x)=e^x$, which leads to the modified ODE $$ u'''''+e^x=e^{-x}(u+e^x)^2, $$ or $$ u'''''=e^{-x}u^2+2u, $$ with boundary conditions $u(0)=u(1)=u'(0)=u'(1)=u''(0)=0$.