Improved Euler Method Notes

Published on: September 29, 2026

This Improved Euler Method Calculator approximates the solution of an initial value problem more accurately than plain Euler, and shows the working line by line. A first slope predicts where the step lands, a second slope is read at that predicted point, and the average of the two is what the step actually uses. Averaging the two slopes is what cancels most of the error. Following the same pattern every time keeps predictor and corrector distinct.

Step-by-step method

  1. Set up the problem.
  2. Write the improved Euler (Heun) update rule.
  3. Apply the predictor and corrector row by row.
  4. Report the final approximation y at x_n.

Formula:

\(y_{k+1} = y_{k} + \tfrac{h}{2}\left(k_{1} + k_{2}\right),\quad k_{1} = f(x_{k}, y_{k}),\; k_{2} = f(x_{k}+h,\, y_{k}+h k_{1})\)
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 improved Euler (Heun) update rule.
  3. Apply the predictor and corrector row by row.
  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 improved Euler (Heun) update rule.

In this problem:

\(y_{k+1} = y_{k} + \tfrac{h}{2}\left(k_{1} + k_{2}\right),\quad k_{1} = f(x_{k}, y_{k}),\; k_{2} = f(x_{k}+h,\, y_{k}+h k_{1})\)

Step 3

Step 3 - Apply the predictor and corrector row by row.

In this problem:

\(\begin{array}{ccc} k & x_k & y_k \\ \hline 0 & 0.0 & 2.0 \\ 1 & 0.2 & 2.42 \\ 2 & 0.4 & 2.8884 \\ 3 & 0.6 & 3.415848 \\ 4 & 0.8 & 4.015335 \\ 5 & 1.0 & 4.702708 \end{array}\)

Step 4

Step 4 - Report the final approximation y at x_n.

In this problem:

\(y(1.0) \approx 4.702708\)

Final Answer

\(y(1.0) \approx 4.702708\)
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 improved Euler (Heun) update rule.
  3. Apply the predictor and corrector row by row.
  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 improved Euler (Heun) update rule.

In this problem:

\(y_{k+1} = y_{k} + \tfrac{h}{2}\left(k_{1} + k_{2}\right),\quad k_{1} = f(x_{k}, y_{k}),\; k_{2} = f(x_{k}+h,\, y_{k}+h k_{1})\)

Step 3

Step 3 - Apply the predictor and corrector row by row.

In this problem:

\(\begin{array}{ccc} k & x_k & y_k \\ \hline 0 & 0.0 & 1.0 \\ 1 & 0.1 & 1.005 \\ 2 & 0.2 & 1.020175 \\ 3 & 0.3 & 1.045986 \end{array}\)

Step 4

Step 4 - Report the final approximation y at x_n.

In this problem:

\(y(0.3) \approx 1.045986\)

Final Answer

\(y(0.3) \approx 1.045986\)
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 improved Euler (Heun) update rule.
  3. Apply the predictor and corrector row by row.
  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 improved Euler (Heun) update rule.

In this problem:

\(y_{k+1} = y_{k} + \tfrac{h}{2}\left(k_{1} + k_{2}\right),\quad k_{1} = f(x_{k}, y_{k}),\; k_{2} = f(x_{k}+h,\, y_{k}+h k_{1})\)

Step 3

Step 3 - Apply the predictor and corrector row by row.

In this problem:

\(\begin{array}{ccc} k & x_k & y_k \\ \hline 0 & 0.0 & 1.0 \\ 1 & 0.1 & 1.11 \\ 2 & 0.2 & 1.24205 \\ 3 & 0.3 & 1.398465 \\ 4 & 0.4 & 1.581804 \end{array}\)

Step 4

Step 4 - Report the final approximation y at x_n.

In this problem:

\(y(0.4) \approx 1.581804\)

Final Answer

\(y(0.4) \approx 1.581804\)
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 improved Euler (Heun) update rule.
  3. Apply the predictor and corrector row by row.
  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 improved Euler (Heun) update rule.

In this problem:

\(y_{k+1} = y_{k} + \tfrac{h}{2}\left(k_{1} + k_{2}\right),\quad k_{1} = f(x_{k}, y_{k}),\; k_{2} = f(x_{k}+h,\, y_{k}+h k_{1})\)

Step 3

Step 3 - Apply the predictor and corrector row by row.

In this problem:

\(\begin{array}{ccc} k & x_k & y_k \\ \hline 0 & 0.0 & 0.0 \\ 1 & 0.5 & 0.25 \\ 2 & 1.0 & 1.0 \\ 3 & 1.5 & 2.25 \\ 4 & 2.0 & 4.0 \end{array}\)

Step 4

Step 4 - Report the final approximation y at x_n.

In this problem:

\(y(2.0) \approx 4.0\)

Final Answer

\(y(2.0) \approx 4.0\)