Numerical Solutions of ODEs: Euler's Method

≈ 20 min

When an initial-value problem $y' = f(x, y),\; y(x_0) = y_0$ has no closed-form solution, Euler's method marches forward in small steps of size $h$ :

y_{n+1} = y_n + h\, f(x_n, y_n), \qquad x_{n+1} = x_n + h.

At each point you use the slope $f(x_n, y_n)$ to take a short straight step, then re-evaluate the slope and repeat.

Worked example. For $y' = y,\; y(0) = 1,\; h = 0.5$ : $y_1 = 1 + 0.5(1) = 1.5$ , then $y_2 = 1.5 + 0.5(1.5) = 2.25.$ The exact answer at $x = 1$ is $e \approx 2.718$ , so Euler underestimates here — a reminder that it is only an approximation.

Apply one Euler step to $y' = y$ with $y(0) = 1$ and step size $h = 0.5$ . Find $y_1$ .

Making the step size $h$ smaller in Euler's method generally…

increases accuracy but needs more steps

decreases accuracy

has no effect on accuracy

always makes the result exact

Continue the previous problem: for $y' = y$ , $y_1 = 1.5$ , $h = 0.5$ , find $y_2$ .

For $y' = x + y$ with $y(0) = 1$ and $h = 0.1$ , take two Euler steps and find $y_2$ .