Calculators / Differential Equations / Euler Method

Euler Method Calculator

Published on: November 23, 2025

This Euler Method Calculator helps you approximate the solution of a first-order differential equation and shows each step clearly. It works by starting from an initial value and repeatedly using the slope from the differential equation together with the chosen step size to estimate the next point. This makes it useful for checking answers, understanding how the Euler method works, and practising differential equations step by step.

Step-by-step method

  1. Rewrite the ODE as y' = f(x,y).
  2. Read the IVP: y(x0) = y0, choose step size h and steps n.
  3. For each step: x_{k+1} = x_k + h.
  4. Compute slope: f_k = f(x_k, y_k).
  5. Update: y_{k+1} = y_k + h f_k.
  6. Report y_n at x_n (and the table if needed).

Formula bank:

Euler: y' = f(x,y)
xk+1 = xk + h
yk+1 = yk + h·f(xk, yk)
fk = f(xk, yk)
yk+1 = yk + h·fk

Example 1: dy/dx = x + y, x, y, y(0)=1, h=0.2, n=4

Step 1 - Write y' = f(x,y).

In this problem: Use the ODE in explicit slope form.

y′ = x + y

Step 2 - Read parameters.

In this problem: Use the initial condition and Euler settings.

y(0) = 1
h = 0.2, n = 4

Step 3 - Euler update (k=0).

In this problem: Compute the slope and update x and y.

x0 = 0, y0 = 1
f0 = 1
x1 = 0 + 0.2 = 0.2
y1 = 1 + 0.2·1 = 1.2

Step 4 - Euler update (k=1).

In this problem: Compute the slope and update x and y.

x1 = 0.2, y1 = 1.2
f1 = 1.4
x2 = 0.2 + 0.2 = 0.4
y2 = 1.2 + 0.2·1.4 = 1.48

Step 5 - Euler update (k=2).

In this problem: Compute the slope and update x and y.

x2 = 0.4, y2 = 1.48
f2 = 1.88
x3 = 0.4 + 0.2 = 0.6
y3 = 1.48 + 0.2·1.88 = 1.856

Step 6 - Euler update (k=3).

In this problem: Compute the slope and update x and y.

x3 = 0.6, y3 = 1.856
f3 = 2.456
x4 = 0.6 + 0.2 = 0.8
y4 = 1.856 + 0.2·2.456 = 2.3472

Step 7 - Summary table.

In this problem: All Euler iterations in one table.

kxkykf(xk,yk)yk+1
00111.2
10.21.21.41.48
20.41.481.881.856
30.61.8562.4562.3472

Step 8 - Final answer.

In this problem: Euler approximation at x_n.

y(xn) ≈ yn = 2.3472(xn = 0.8)

Final answer: Approximation

Example 2: dy/dx = y - x^2 + 1, x, y, y(0)=0.5, h=0.2, n=5

Step 1 - Write y' = f(x,y).

In this problem: Use the ODE in explicit slope form.

y′ = 1 + y − x2

Step 2 - Read parameters.

In this problem: Use the initial condition and Euler settings.

y(0) = 0.5
h = 0.2, n = 5

Step 3 - Euler update (k=0).

In this problem: Compute the slope and update x and y.

x0 = 0, y0 = 0.5
f0 = 1.5
x1 = 0 + 0.2 = 0.2
y1 = 0.5 + 0.2·1.5 = 0.8

Step 4 - Euler update (k=1).

In this problem: Compute the slope and update x and y.

x1 = 0.2, y1 = 0.8
f1 = 1.76
x2 = 0.2 + 0.2 = 0.4
y2 = 0.8 + 0.2·1.76 = 1.152

Step 5 - Euler update (k=2).

In this problem: Compute the slope and update x and y.

x2 = 0.4, y2 = 1.152
f2 = 1.992
x3 = 0.4 + 0.2 = 0.6
y3 = 1.152 + 0.2·1.992 = 1.5504

Step 6 - Euler update (k=3).

In this problem: Compute the slope and update x and y.

x3 = 0.6, y3 = 1.5504
f3 = 2.1904
x4 = 0.6 + 0.2 = 0.8
y4 = 1.5504 + 0.2·2.1904 = 1.98848

Step 7 - Euler update (k=4).

In this problem: Compute the slope and update x and y.

x4 = 0.8, y4 = 1.98848
f4 = 2.34848
x5 = 0.8 + 0.2 = 1
y5 = 1.98848 + 0.2·2.34848 = 2.458176

Step 8 - Summary table.

In this problem: All Euler iterations in one table.

kxkykf(xk,yk)yk+1
000.51.50.8
10.20.81.761.152
20.41.1521.9921.5504
30.61.55042.19041.98848
40.81.988482.348482.458176

Step 9 - Final answer.

In this problem: Euler approximation at x_n.

y(xn) ≈ yn = 2.458176(xn = 1)

Final answer: Approximation