5.18 Second order differential equationsIB Maths: Applications and Interpretation HL: Revision notes

A second-order differential equation contains $\frac{d^2x}{dt^2}$, the acceleration when $x$ is a displacement. They arise from Newton's second law. A mass $m$ on a spring with stiffness $k$ gives $m\frac{d^2x}{dt^2}=-kx$, so $\frac{d^2x}{dt^2}=-\frac{k}{m}x$. Adding a damping (friction) force proportional to velocity gives $\frac{d^2x}{dt^2}+a\frac{dx}{dt}+bx=0$. A pendulum gives $\frac{d^2\theta}{dt^2}=-\frac{g}{L}\sin\theta$. You are not asked to derive these equations: in examinations the equation is given. You need to know how to solve it numerically and how to analyse it with a phase portrait.

Euler's method needs first-order equations, so introduce a second variable for the velocity. For $\frac{d^2x}{dt^2}=f\left(x,\frac{dx}{dt},t\right)$ let $y=\frac{dx}{dt}$. Then $\frac{dx}{dt}=y,\qquad \frac{dy}{dt}=f(x,y,t).$ This is a coupled system in $x$ and $y$. Example: $\frac{d^2x}{dt^2}=-4x-0.5\frac{dx}{dt}$ becomes $\frac{dx}{dt}=y$ and $\frac{dy}{dt}=-4x-0.5y$. The initial conditions become $x(0)$ and $y(0)=\frac{dx}{dt}(0)$.

Apply Euler's method to both equations, using the old values on the right: $t_{n+1}=t_n+h,\quad x_{n+1}=x_n+h\,y_n,\quad y_{n+1}=y_n+h\,f(x_n,y_n,t_n).$ Example: $\frac{d^2x}{dt^2}=-2x+t$, $x(0)=1$, $\frac{dx}{dt}(0)=0$, $h=0.1$. Step 1 (using $t=0$): $x_1=1+0.1(0)=1$ and $y_1=0+0.1(-2(1)+0)=-0.2$. Step 2 (using $t=0.1$): $x_2=1+0.1(-0.2)=0.98$ and $y_2=-0.2+0.1(-2(1)+0.1)=-0.39$. So at $t=0.2$, $x\approx0.98$ and $\frac{dx}{dt}\approx-0.39$. Use a spreadsheet for more steps, and a GDC in radian mode if the equation contains $\sin x$. Euler's method slowly drifts: for $\frac{d^2x}{dt^2}=-x$, each step multiplies $x^2+y^2$ by $(1+h^2)$, although the true value is constant.

For $\frac{d^2x}{dt^2}+a\frac{dx}{dt}+bx=0$, the coupled system is $\frac{dx}{dt}=y$ and $\frac{dy}{dt}=-bx-ay$, with matrix $M=\begin{pmatrix} 0 & 1 \\ -b & -a \end{pmatrix}$. Its eigenvalues satisfy $\lambda^2+a\lambda+b=0.$ Then use the phase portrait method from 5.17, with $x$ (position) and $y$ (velocity) as the axes. Example: $\frac{d^2x}{dt^2}+2\frac{dx}{dt}+10x=0$ has $\lambda=-1\pm3i$.

• $a=0$, $b>0$: imaginary eigenvalues $\pm i\sqrt{b}$, so closed ellipses: an undamped oscillation of constant amplitude.
• Complex eigenvalues with negative real part: a spiral towards the origin, a damped oscillation whose amplitude decays.
• Two distinct negative real eigenvalues: a stable node, so $x$ returns to $0$ without oscillating (heavy damping).
• $b<0$: the eigenvalues are real with opposite signs, a saddle point, so $x$ runs away (an unstable situation). The origin $x=0,\ y=0$ is the rest position: the mass is at the centre with no velocity.

Euler's method gives numbers: the position and velocity at particular times, for any given equation, including non-linear ones like the pendulum. The phase portrait gives the qualitative behaviour of linear equations: whether solutions grow, decay or cycle, without calculating each value. For a non-linear equation such as $\frac{d^2\theta}{dt^2}=-10\sin\theta$, replacing $\sin\theta$ by $\theta$ for small angles gives a linear equation that you can analyse with a phase portrait. In a question, check which is asked: 'estimate $x$ when $t=\ldots$' needs Euler's method, while 'describe the long-term behaviour' needs the eigenvalues.