How to solve a 4th order ODE with given boundary conditions?

[seang]

Homework Statement

Hello, I should probably know how to do this, but I am confused as to how to solve the following 4th order ODE:

\begin{align}<br /> &amp; EI \frac{\mathrm{d}^4 w}{\mathrm{d} x^4} = 0 \\<br /> &amp; w|_{x = 0} = 0 \quad ; \quad \frac{\mathrm{d} w}{\mathrm{d} x}\bigg|_{x = 0} = 0 \quad ; \quad<br /> \frac{\mathrm{d}^2 w}{\mathrm{d} x^2}\bigg|_{x = L} = 0 \quad ; \quad -EI \frac{\mathrm{d}^3 w}{\mathrm{d} x^3}\bigg|_{x = L} = F\,<br /> \end{align}<br />

The well-known solution is:

w = \frac{F}{6 EI}(3 L x^2 - x^3)\,~.

...but I don't know how to obtain it myself.

The Attempt at a Solution

Since all the roots of the characteristic equation would be 0, the solution should be:

w = c1*exp(0*x) + c2*exp(0*x) +...+c4*exp(0*x)

Then normally one would use the initial conditions to get the constants, but that gives sth like the following system:

c1+c2+c3+c4 = 0
0 = 0
0 = 0
0 = 0

haha

in fact, I am not sure how one could get an equation with powers of x solving the equation this way. I must be going about this wrong or making a very simple mistake somewhere...

 

[ehild]

Integrate the ODE step by step. d⁴w/dx⁴=0, what do you know about d³w/dx³?

ehild

 

[seang]

wow, so simple. thanks a lot.

re-reading my ODE book, it says that the method i was using assumes the answer is exponentials. i guess i should read more carefully