# Thread: Second order ODE with general forcing function of two variables

1. ## Second order ODE with general forcing function of two variables

Hi,

I'm trying to solve for the modal transient dynamic response of a beam subjected to arbitrary loading in $\displaystyle t$ (time) and $\displaystyle x$ (measurement along the length of the beam). I've already solved for the eignefunctions that describe the beams displacement. I'm now trying to solve for the temporal solution. I get to

$\displaystyle \ddot{T_{n}}(t) + T_{n}(t) = R(x,t)$

where $\displaystyle R(x,t)$ is an arbitrary function determined from some integration and operation on the forcing fuction.

Question: Does anyone know how to obtain a solution to this equation for $\displaystyle T_{n}(t)$? I'm not sure what to use as an initial solution guess.

Any help is much appreciated.

Thanks.

2. That's just the standard $\displaystyle y''+y=f(t)$ which in your case is $\displaystyle T''+T=f(t,x)$ and that has the general solution:

$\displaystyle T(t)=c_1\cos(t)+c_2\sin(t)+\int_{a}^t f(\beta,x)\sin(t-\beta)d\beta$

assuming $\displaystyle f(t,x)$ is integrable in the interval $\displaystyle a<t<b$. That solution is via variation of parameter

3. Hi Shawsend,

I went through my old ODE book and found variation of parameters. Thanks for the guidance. I'll give this a shot.