4nd order differential eq

[pgioun]

Hi,
I want to solve the Euler-Bernoulli eq numerically using a c++ library.

EI $y^{4}$(x)=f(x), y(0)=0,y'(0)=0,y(L)=0,y'(L)=0.

where L is the length of the beam and the initial conditions are for a cantilever.

In order to achieve that I have to make it a set of 1st ode.
How this system of 1st order ode would be like and how the initial conditions
should be rearranged?

How should the shooting method be applied to this system of 4 odes problem?

Thanks

 

[Mute]

To turn a higher order ODE into a system of first order ODEs you just define new variables which are equal to derivatives of the variable you want to solve for.

Here's an example with a second order ODE:

$$y''(x) + y'(x) - y (x) = f(x)$$

To make this a system of first order ODEs, define $u(x)= y'(x)$. Then, it immediately follows that $u'(x) = y''(x) = -y'(x) + y(x) = -u(x) + y(x)$. The system of equations is thus

$$\begin{eqnarray*}
y'(x) & = & u(x) \\
u'(x) & = & -u(x) + y(x)
\end{eqnarray*}$$

This is generally how you want your system of ODEs to look: $v_i'(x) = f_i(x,v_1(x),v_2(x),\dots,v_i(x),\dots,v_n(x))$. In the example above, n = 2 and $v_1 = y,~v_2 = u$. In your case, n = 4.

If this problem had initial conditions $y(0) = 0,~y'(0)=1$, this would correspond to $y(0)=0,u(0)=1$.

Now try it with your ODE.