# Thread: Help on solving second order equation with constant

1. ## Help on solving second order equation with constant

Hi,

I am hoping someone can give me a nudge in the right direction on understanding and solving the following equations:

Au''(t)+Bu'(t)-Cu(t)+Dp(t)+F=G(t)

-Du'(t) + Hp'(t) = -I(t)

where u(t) and p(t) are functions of time. I have taken diff eq classes several years ago, and am refreshing my memory but would like an expert to point out anything unusual. The F is a constant and not associated with any time function.
Thanks!

2. Well, one interesting feature is that, assuming u and p are the unknown functions for which you wish to solve, and G and I are known functions, then one of your equations can be integrated directly to produce

$\displaystyle-Du(t)+Hp(t)=-\int I(t)\,dt:=i(t),$ from which we may solve for

$Dp(t)=\dfrac{Di(t)+D^{2}u}{H}.$

Hence, the first ODE becomes

$A\ddot{u}+B\dot{u}-Cu+\dfrac{Di(t)+D^{2}u}{H}=G(t)-F,$ or

$\displaystyle A\ddot{u}+B\dot{u}+\left(\frac{D^{2}}{H}-C\right)u=G(t)-F-\dfrac{Di(t)}{H}.$

Now you have a regular ol' inhomogeneous second-order ODE with constant coefficients. Use your favorite method here.

3. Thank you more than I can express! I am going to try and get this into MATLAB.

4. You're welcome! Let me know if you have any further difficulties.

5. $\displaystyle-Du(t)+Hp(t)=-\int I(t)\,dt:=i(t),$

Okay, now that the OP has got it I can ask one of my own. I had gotten to the point of integrating the second equation to find p(t) as a function of u(t). But is there still a solution if I(t) is not integrable?

-Dan

6. Hmm. Well, you might be able to work with an eigenvalue approach, where the integrability of $I(t)$ is never assumed. If you let $x_{1}(t)=u(t), x_{2}(t)=\dot{u}(t),$ and $x_{3}(t)=p(t),$ then you get the following system (it just works better with three coordinates than four):

$\displaystyle\frac{d}{dt}\begin{bmatrix}x_{1}\\x_{ 2}\\x_{3}\end{bmatrix}=\begin{bmatrix}0 &1 &0\\C/A &-B/A &-D/A\\0 &D/H &0\end{bmatrix}\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\ end{bmatrix}+\begin{bmatrix}0\\(G(t)-F)/A\\-I(t)/H\end{bmatrix}.$

You could use the usual eigenvalue procedure to find the homogeneous solution, and then use undetermined coefficients to find a particular solution, perhaps.