A problem from Dennis Zill's book

Question

I have the following exercises from Dennis Zill's book, anyone can give me some help to resolve this exercises, because with my classmate, How I find the solution of this problem?, Regards!

"A cantilever beam of length L is embedded at its right end, and a horizontal tensile force of P pounds is applied to its free left end. When the origin is taken at its free end, the deflection y(x) of the beam can be shown to satisfy the differential equation $$Ely''=Py-w(x)\dfrac{x}{2}$$ Find the deflection of the cantilever beam if $w(x)=w_{0}x$, $0<x<L$, and $y(0)=0$, $y'(L)=0$".

Answer 1

A particular solution of the complete equation is $$ y={w_0\over2P}x^2+{El\over P^2}w_0, $$ add to this the general solution of the homogeneous equation.

Answer 2

Write $y''-\frac{P}{El}y=\frac{-w(x)}{El}\frac{x}{2}$ to see that you have a linear, non-homogeneous second order differential equation which can be solved by taking

$y=y_h+y_p$ where $y_h$ is the general solution to the homogeneous case and $y_p$ is any other particular solution to the original equation.

Before substituting $w(x)=w_0x$ though, you need to be careful because as it stands, $w$ is not even defined at $x=0$ or $x=L$. You need to extend $w$ so that it is defined at these points. This is easy of course.

The real problem here is that the conditions you are given are NOT initial conditions; one of them is a boundary condition. Therefore, the theorem on uniqueness of solution does not apply, which means that any solution you do get may not be the only one, if indeed you happen to find one.

I imagine Zill deals with these difficulties in his text.