what is the best way to solve a partial differential equation: $$ (1-ax)(∂^4 y)/(∂x^4)+2a (∂^3 y)/(∂x^3)=0 $$
like in ordinary differential equations I tried the power series method (I'm not very good with differential equations). I got something like: $$y= C_1+C_2+C_3 (1+(1/3) ax)+C_4 (1-ax)$$
which is difficult subjecting to the boundary conditions:
$$ y=0,y''=0,x=0 \\ y=M,y''=(-1-y')/k(1-ax),x=z $$
Can anyone help?