# Solution of 4th order ODE

1. Aug 7, 2010

### seang

1. The problem statement, all variables and given/known data

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} & EI \frac{\mathrm{d}^4 w}{\mathrm{d} x^4} = 0 \\ & w|_{x = 0} = 0 \quad ; \quad \frac{\mathrm{d} w}{\mathrm{d} x}\bigg|_{x = 0} = 0 \quad ; \quad \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\, \end{align}

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.

3. 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...

2. Aug 7, 2010

### ehild

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

ehild

3. Aug 7, 2010

### 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