How Do You Solve a Second-Order Differential Equation Using Euler's Method?

Convert the second-order differential equation into two coupled first-order equations, then apply Euler's method to both equations simultaneously at every step. This is HL-only content in AHL 5.18: Euler's method for second-order differential equations.

For an equation of the form

$$\frac{d^2x}{dt^2}=f\left(x,\frac{dx}{dt},t\right),$$

introduce the velocity variable

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

Therefore, $\frac{dx}{dt}=v$, and the original equation becomes the coupled system

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

At step $n$, calculate both new values from the same old state. With step size $h$, the recurrence relations are

$$x_{n+1}=x_n+hv_n,$$ $$v_{n+1}=v_n+h f(x_n,v_n,t_n),$$ $$t_{n+1}=t_n+h.$$

Both derivatives are evaluated using $x_n$, $v_n$, and $t_n$ before either variable is updated.

This simultaneous update matters because Euler's method replaces each derivative by the gradient at the start of the interval. Geometrically, the current state determines one tangent step in position and one tangent step in velocity; only after both estimates are found does the method advance to the next row.

Worked Example

Consider

$$\frac{d^2x}{dt^2}=-x, \qquad x(0)=1, \qquad x'(0)=0,$$

with $h=0.1$. Let $v=x'$, giving $x'=v$ and . The initial state is .

Thus, Euler's method estimates $x(0.2)\approx0.99$. Each ordered pair $(x_n,v_n)$ approximates both the position and its rate of change at time . A smaller step size usually improves accuracy because the tangent-line approximation is applied over shorter intervals, although more iterations are required.

For organized working, create columns for $t_n$, $x_n$, $v_n$, , and . This makes the old-state substitutions visible and reduces update errors. On a GDC or spreadsheet, enter the recurrences once, fill them downward, and retain unrounded values.

A common misconception is to use the newly calculated $x_{n+1}$ when finding $v_{n+1}$. This is incorrect: both updates must use the old values from row $n$, not a mixture of old and new values.

Exam technique: In IB Math AI HL, define the new variable, state both first-order equations, write both recurrences, and show the first iteration explicitly. Keep full GDC precision throughout and round only the final estimate, normally to three significant figures unless instructed otherwise.