# Coupled 2nd Order DE: Convert to system of 1st Order

It's been a while since I've played with systems of ODEs, and I seem to have forgotten some of the tricks. As an example, I have two coupled nonlinear DE that I want to convert to a system of four 1st order nonlinear DE. But, the normal way of making variable substitutions is not working of me.

Two 2nd order nonlinear DE:
##\ddot{\alpha} + \ddot{\beta}cos(\alpha-\beta)+\dot{\beta}^2sin(\alpha-\beta)=\tau_{1}##
##\ddot{\beta} + \ddot{\alpha}cos(\alpha-\beta)-\dot{\alpha}^2sin(\alpha-\beta)=\tau_{2}##

If I substitute variables ##x_1=\alpha##, ##x_2=\dot{\alpha}##,##x_3=\beta##,##x_4=\dot{\beta}##, like my initial reaction was, I get:

##\dot{x_1}=x_2##
##\dot{x_2}=-\dot{x_4}cos(x_1-x_3)-x_4^2sin(x_1-x_3)+\tau_1##
##\dot{x_3}=x_4##
##\dot{x_4}=-\dot{x_2}cos(x_1-x_3)+x_2^2sin(x_1-x_3)+\tau_2##

which isn't quite right, since it is not in the form ##\dot{\vec{\textbf{x}}}=\vec{\textbf{f}}(t,\vec{\textbf{x}})##

I'm trying to get it a form where I can run it through ODE45 or a similar solver.

Refresh my memory! :)

EDIT! I believe I figured it out. I did do it right, just implementing it in Matlab had me confused. But, all I have to do in my function is use the old values for ##\dot{x_2}## and ##\dot{x_4}##. Seeing those derivatives in the rhs just threw me off. Is this the right way to go about it?

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AlephZero
Homework Helper
You can write your equations as
$$\begin{bmatrix} 1 & \cos(\alpha - \beta) \\ \cos(\alpha - \beta) & 1 \end{bmatrix} \begin{bmatrix} \ddot\alpha \\ \ddot\beta \end{bmatrix} = \begin{bmatrix} -\dot\beta^2 \sin(\alpha - \beta) + r_1 \\ \dot\alpha^2 \sin(\alpha - \beta) + r_2\end{bmatrix}$$
So
$$\begin{bmatrix} \ddot\alpha \\ \ddot\beta \end{bmatrix} = \begin{bmatrix} 1 & \cos(\alpha - \beta) \\ \cos(\alpha - \beta) & 1 \end{bmatrix}^{-1} \begin{bmatrix} -\dot\beta^2 \sin(\alpha - \beta) + r_1 \\ \dot\alpha^2 \sin(\alpha - \beta) + r_2\end{bmatrix}$$

The matrix is singular if ##\alpha = \beta##, but in that case you only have one differential equation not two.

You can write your equations as
$$\begin{bmatrix} 1 & \cos(\alpha - \beta) \\ \cos(\alpha - \beta) & 1 \end{bmatrix} \begin{bmatrix} \ddot\alpha \\ \ddot\beta \end{bmatrix} = \begin{bmatrix} -\dot\beta^2 \sin(\alpha - \beta) + r_1 \\ \dot\alpha^2 \sin(\alpha - \beta) + r_2\end{bmatrix}$$
So
$$\begin{bmatrix} \ddot\alpha \\ \ddot\beta \end{bmatrix} = \begin{bmatrix} 1 & \cos(\alpha - \beta) \\ \cos(\alpha - \beta) & 1 \end{bmatrix}^{-1} \begin{bmatrix} -\dot\beta^2 \sin(\alpha - \beta) + r_1 \\ \dot\alpha^2 \sin(\alpha - \beta) + r_2\end{bmatrix}$$

The matrix is singular if ##\alpha = \beta##, but in that case you only have one differential equation not two.

That makes sense, but is that the only way to implement it into Matlab (ode45)? I thought I was onto something (see my edit in my original post), but now I'm thinking it won't work.

Chestermiller
Mentor
Move everything with overdots to the left hand sides. Then re-express the left hand side as a matrix times the column vector of the dotted first derivatives. So the equation is

Mxdot=f(x,t)

In your subroutine for evaluating derivatives, first calculate the right hand side vector. Then premultiply it by the inverse of M. Some equation solvers accept the more general inputs as M and f(x,t), and do the inversion for you. I'm not sure whether the equation solver you are using will do this.

Chet

Move everything with overdots to the left hand sides. Then re-express the left hand side as a matrix times the column vector of the dotted first derivatives. So the equation is

Mxdot=f(x,t)

In your subroutine for evaluating derivatives, first calculate the right hand side vector. Then premultiply it by the inverse of M. Some equation solvers accept the more general inputs as M and f(x,t), and do the inversion for you. I'm not sure whether the equation solver you are using will do this.

Chet

Thanks, Chet. Turns out I was able to keep it in my original form (system of implicit DEs) and use Matlabs ode15i solver. I believe it does a similar operation by inverting the matrix....but it just saves me a lot of work! It wouldn't be too bad, but when the system I have has 8 state variables, it gets messy really fast.

Anyways, looks like ode15i was the solution I was looking for.