How Do You Convert a Second-Order Differential Equation Into a Coupled First-Order System?

Introduce a new variable equal to the first derivative. This replaces one second-order differential equation with two coupled first-order differential equations, which can then be solved numerically using Euler's method.

The General Method

Suppose the equation is written as

$$\frac{d^2y}{dt^2}=f\left(t,y,\frac{dy}{dt}\right).$$

Define a new state variable

$$v=\frac{dy}{dt}.$$

This immediately gives the first equation:

$$\frac{dy}{dt}=v.$$

Since $v$ is the first derivative of $y$,

$$\frac{dv}{dt}=\frac{d^2y}{dt^2}=f(t,y,v).$$

Therefore, the coupled system is

$$\frac{dy}{dt}=v, \qquad \frac{dv}{dt}=f(t,y,v).$$

The equations are coupled because each update may depend on both state variables. Rearranging first is essential because the second Euler update needs an explicit expression for the highest derivative. The substitution does not approximate the model; it only rewrites the same information in state-space form.

The state variables have distinct roles. $y$ is the original quantity, while $v$ records its rate of change. The first equation links the variables; the second describes how the rate changes. Writing both equations explicitly is important because Euler's method updates both components at every step.

Worked Example

Convert

$$\frac{d^2y}{dt^2}+3\frac{dy}{dt}-2y=\sin t$$

into a coupled first-order system. Rearranging gives

$$\frac{d^2y}{dt^2}=\sin t-3\frac{dy}{dt}+2y.$$

Let $v=\frac{dy}{dt}$. The system is

$$\frac{dy}{dt}=v, \qquad \frac{dv}{dt}=\sin t-3v+2y.$$

If $y(0)=4$ and $y'(0)=-1$, the initial conditions become $y(0)=4$ and .

For step size $h$, Euler's method gives

$$y_{n+1}=y_n+hv_n,$$ $$v_{n+1}=v_n+h f(t_n,y_n,v_n), \qquad t_{n+1}=t_n+h.$$

For example, with $h=0.1$, the initial acceleration is $\sin 0-3(-1)+2(4)=11$. Hence the first update gives $y_1=$ and . Both calculations must use the original values and , not a partially updated state.

At each iteration, calculate both new values from the same old state. Updating $y$ first and then using $y_{n+1}$ to calculate $v_{n+1}$ is not standard Euler's method and changes the numerical scheme. A table can help organize the first iteration clearly.

A common misconception is to define $v=y$. This does not reduce the order; the new variable must equal the first derivative.

IB Exam Technique

This is HL-only content from AHL 5.18. Examiners expect the substitution, both first-order equations, converted initial conditions, and the first complete Euler iteration.