# How do I solve this Differential Algebraic Equation?

I am trying to model a physical system which is goverened by the equation:

$$(r(t))^2 \cdot \frac{d^2 r(t)}{dt^2} = c q(t)$$

I am a junior electrical engineering student, so I have had courses in linear algebra, ordinary differential equations, and LTI system theory + Fourier/Laplace/DFT/Z transforms.

Based on what I can tell about this equation, it is a Differential Algebraic Equation, which I am unfamiliar with how to solve. I would like to solve for $r(t)$ given an arbitrary function of $q(t)$ - say a sinusoidal, exponential, or polynomial function.

From what I've read online, an analytic solution may not be possible, so numerical solutions are fine too.

Also, all I can find online is how to solve a SYSTEM of DAEs, not simply a singular DAE equation.

What I would like to do is to be able to predict the output $r(t)$ given some input signal $q(t)$

• It's not a Differential Algebraic Equation, just a differential equation. – Robert Israel May 31 '18 at 18:34

You can use a Runge-Kutta solver, for instance, but as your equation is of the second order, you will turn it to a system anyway:

$$\begin{cases}\dfrac{dr(t)}{dt}=s(t),\\\dfrac{ds(t)}{dt}=c\dfrac{q(t)}{r^2(t)}.\end{cases}$$

• Could you show how you arrived at that system of equations? – rhm May 31 '18 at 17:21
• @rhm: it is trivial, eliminate $s$ to see. – Yves Daoust May 31 '18 at 17:43
• Let $s=\dfrac{dr}{dt}$ then $\dfrac{ds}{dt}=\dfrac{d^2r}{dt^2}$ – N8tron May 31 '18 at 17:54
• Many solvers (e.g. Maple's dsolve(..., numeric)) will accept a second-order DE without turning it into a system. – Robert Israel May 31 '18 at 18:43