Systems of ODEs

Many practical problems in engineering and science require the solution of a system of simultaneous ordinary differential equations, such as:

$$\begin{array}{rl}\frac{d{y}_1}{dx}& =f_1(x,y_1,y_2,\dots ,y_n)\\ \frac{d{y}_2}{dx}& =f_2(x,y_1,y_2,\dots ,y_n)\\ & ⋮\\ \frac{d{y}_n}{dx}& =f_n(x,y_1,y_2,\dots ,y_n)\end{array}$$

This requires that $n$ initial conditions be known at the starting value of $x$.

The one-step methods we have investigated can be applied to solve systems (Euler’s Method, Heun’s Method, Runge-Kutta Method). We apply the one-step method for every equation, at each step, before proceeding to the next step.

Example

Solve the following set of differential equations using Euler’s method, assuming that $x=0,y_1=4,y_2=6$. Integrate to $x=2$ with a step size of $0.5$.

$$\frac{d{y}_1}{dx}=-0.5y_1,\frac{d{y}_2}{dx}=4-0.3y_2-0.1y_1$$

Recall:

$$y_{i+1}=y_i+f(x_i,y_i)h$$

Thus, we have:

$$\begin{array}{rl}{y}_1(0)& =4\\ y_1(0.5)& =4+[-0.5(4)]0.5=3\\ & \\ y_2(0)& =6\\ y_2(0.5)& =6+[4-0.3(6)-0.1(4)]0.5=6.9\end{array}$$

Note that $y_1(0)=4$ is used in the second equation rather than $y_1(0.5)=3$ computed with the first equation. We are using consistent values from the current step, to calculate the function values at the next step.

Continuing the process: