4th order linear, with constant piecewise coefficients

[sir_manning]

Hi everyone

I'm modeling the dynamics of a cantilever that has a non-constant linear density profile, i.e.

$$\rho(x)=\rho_{1} \0 \leq x \leq x_{0}$$
$$\rho(x)=\rho_{2} \0 x_{0} \leq x \leq l$$
$$\rho(x)=0 \0$$ otherwise

My differential equation is:

$$\frac{ d^4 \phi(x) } {d x^4} = \phi(x) \rho(x)$$

I'm wondering what tools I should through at this thing. I was thinking Fourier transforms, so I re-wrote $$\rho(x)$$ as the difference between two box functions. However, when I take the transform I have the convolution of $$\Phi(x)$$, which is unknown, with some $$sinc$$ functions.

Could someone point me in the right direction for how to tackle an equation like this?

 

[AlephZero]

If you want an analytic solution, you could solve the ODE for the two parts separately, and then equate $\phi$ and the first 3 derivatives to eliminate 4 of the 8 arbitary constants.

A numerical solution might be easier. This sort of thing is trivial to model in finite element analysis.