Solving a first order differential equation by fourth order Runge-Kutta







Numerical Methods

Solving a first order differential equation by fourth order Runge-Kutta

Any first order differential equation of the form,

\[ \begin{equation} \frac{dx}{dt} = f(x,t) \end{equation} \]

can be solved numerically using a fourth order Runge-Kutta routine. The code that solves this equation for $f(x,t) = -x + \sin (t)$ and the intial condition $x(t=0)=0$ is shown below.

<script> var x=0; var t=0; var dt=0.01; function f(x,t) { dxdt = -x + Math.sin(t); return dxdt; } for (j=0; j<20; j++){ document.write("t="+t+" x="+x+"<\/br>"); k1 = dtf(x,t); k2 = dtf(x+0.5k1,t+0.5dt); k3 = dtf(x+0.5k2,t+0.5dt); k4 = dtf(x+k3,t+dt); x += (k1+2k2+2k3+k4)/6; t += dt; } </script>