AHL 5.18 (HL)—Second-order differential equations

For $d^2x/dt^2=f(x,dx/dt,t)$, define $y=dx/dt$. Then rewrite the equation as the coupled system $dx/dt=y$ and $dy/dt=f(x,y,t)$.

With step $h$, update simultaneously from the current state: $x_{n+1}=x_n+hy_n$ and $y_{n+1}=y_n+h f(x_n,y_n,t_n)$. Initial position and velocity provide the two starting values.

Example

For $x''=-x$, $x(0)=1$, $x'(0)=0$ and $h=0.1$, set $y=x'$. One Euler step gives $x_1=1+0.1(0)=1$ and $y_1=0+0.1(-1)=-0.1$.

The syllabus requires numerical Euler and phase-portrait treatment, not a general characteristic-equation solution. Use both current values in one update and do not confuse velocity $y$ with position $x$.