[Math] Techniques for solving coupled differential equations

Question

I am trying to solve a system of coupled differential equations to plot streamlines using Matlab.

The equations are these:

\begin{align}
\frac{\mathrm dx}{\mathrm dt} &= -3x -5y \\
\frac{\mathrm dy}{\mathrm dt} &= 5x + 3y
\end{align}

What method do you suggest for solving this system? I'd greatly appreciate any insight or suggestion. No need to solve the system 🙂 as long as you tell me what literature I can refer to.

Thanks!

Answer 1

Another approach:

Consider the following IVP problem:

\begin{align} \frac{\mathrm{d}x}{\mathrm{d}t} &= -3x - 5y \\ \frac{\mathrm{d}y}{\mathrm{d}t} &= 5x + 3y \end{align}

with $x(0)=x_0$ and $y(0)=y_0$.

Then, Laplace-transform both sides of both equations to get:

\begin{align} s X(s) - x_0 & = - 3 X(s) -5 Y(s) \\ s Y(s) - y_0 & = 5 X(s) + 3Y(s) , \end{align}

which is an algebraic system for $X(s) = \mathcal{L}_sx(t)$ and $Y(s) = \mathcal{L}_sy(t)$. Solve for the unknowns using (for example) Gauss elimination and compute the inverse Laplace transfrom to get the solution.

Cheers!