Euler's Method Notes

Published on: September 29, 2026

This Euler Method Calculator approximates the solution of an initial value problem step by step, and shows the working line by line. At each point the slope given by the equation is treated as constant across one step, so the curve is followed by a short straight line. Repeating that from the starting point walks the approximation forward. Following the same pattern every time makes it clear how the error grows with the step size.

Step-by-step method

  1. Set up the problem.
  2. Write the Euler update rule.
  3. Apply the rule row by row, updating x and y each step.
  4. Report the final approximation y at x_n.

Formula:

\(y_{k+1} = y_{k} + h \, f(x_{k}, y_{k})\)
See Example 1 Hide Example 1

Problem

\(y' = - x + y,\quad y(0.0) = 2.0,\; h = 0.2,\; n = 5\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the Euler update rule.
  3. Apply the rule row by row, updating x and y each step.
  4. Report the final approximation y at x_n.

Step 1

Step 1 - Set up the problem.

In this problem:

\(y' = - x + y,\quad y(0.0) = 2.0,\; h = 0.2,\; n = 5\)

Step 2

Step 2 - Write the Euler update rule.

In this problem:

\(y_{k+1} = y_{k} + h \, f(x_{k}, y_{k})\)

Step 3

Step 3 - Apply the rule row by row, updating x and y each step.

In this problem:

\(\begin{array}{ccc} k & x_k & y_k \\ \hline 0 & 0.0 & 2.0 \\ 1 & 0.2 & 2.4 \\ 2 & 0.4 & 2.84 \\ 3 & 0.6 & 3.328 \\ 4 & 0.8 & 3.8736 \\ 5 & 1.0 & 4.48832 \end{array}\)

Step 4

Step 4 - Report the final approximation y at x_n.

In this problem:

\(y(1.0) \approx 4.48832\)

Final Answer

\(y(1.0) \approx 4.48832\)
See Example 2 Hide Example 2

Problem

\(y' = x y,\quad y(0.0) = 1.0,\; h = 0.1,\; n = 3\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the Euler update rule.
  3. Apply the rule row by row, updating x and y each step.
  4. Report the final approximation y at x_n.

Step 1

Step 1 - Set up the problem.

In this problem:

\(y' = x y,\quad y(0.0) = 1.0,\; h = 0.1,\; n = 3\)

Step 2

Step 2 - Write the Euler update rule.

In this problem:

\(y_{k+1} = y_{k} + h \, f(x_{k}, y_{k})\)

Step 3

Step 3 - Apply the rule row by row, updating x and y each step.

In this problem:

\(\begin{array}{ccc} k & x_k & y_k \\ \hline 0 & 0.0 & 1.0 \\ 1 & 0.1 & 1.0 \\ 2 & 0.2 & 1.01 \\ 3 & 0.3 & 1.0302 \end{array}\)

Step 4

Step 4 - Report the final approximation y at x_n.

In this problem:

\(y(0.3) \approx 1.0302\)

Final Answer

\(y(0.3) \approx 1.0302\)
See Example 3 Hide Example 3

Problem

\(y' = x + y,\quad y(0.0) = 1.0,\; h = 0.1,\; n = 4\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the Euler update rule.
  3. Apply the rule row by row, updating x and y each step.
  4. Report the final approximation y at x_n.

Step 1

Step 1 - Set up the problem.

In this problem:

\(y' = x + y,\quad y(0.0) = 1.0,\; h = 0.1,\; n = 4\)

Step 2

Step 2 - Write the Euler update rule.

In this problem:

\(y_{k+1} = y_{k} + h \, f(x_{k}, y_{k})\)

Step 3

Step 3 - Apply the rule row by row, updating x and y each step.

In this problem:

\(\begin{array}{ccc} k & x_k & y_k \\ \hline 0 & 0.0 & 1.0 \\ 1 & 0.1 & 1.1 \\ 2 & 0.2 & 1.22 \\ 3 & 0.3 & 1.362 \\ 4 & 0.4 & 1.5282 \end{array}\)

Step 4

Step 4 - Report the final approximation y at x_n.

In this problem:

\(y(0.4) \approx 1.5282\)

Final Answer

\(y(0.4) \approx 1.5282\)
See Example 4 Hide Example 4

Problem

\(y' = 2 x,\quad y(0.0) = 0.0,\; h = 0.5,\; n = 4\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the Euler update rule.
  3. Apply the rule row by row, updating x and y each step.
  4. Report the final approximation y at x_n.

Step 1

Step 1 - Set up the problem.

In this problem:

\(y' = 2 x,\quad y(0.0) = 0.0,\; h = 0.5,\; n = 4\)

Step 2

Step 2 - Write the Euler update rule.

In this problem:

\(y_{k+1} = y_{k} + h \, f(x_{k}, y_{k})\)

Step 3

Step 3 - Apply the rule row by row, updating x and y each step.

In this problem:

\(\begin{array}{ccc} k & x_k & y_k \\ \hline 0 & 0.0 & 0.0 \\ 1 & 0.5 & 0.0 \\ 2 & 1.0 & 0.5 \\ 3 & 1.5 & 1.5 \\ 4 & 2.0 & 3.0 \end{array}\)

Step 4

Step 4 - Report the final approximation y at x_n.

In this problem:

\(y(2.0) \approx 3.0\)

Final Answer

\(y(2.0) \approx 3.0\)