Midpoint Method Calculator

Calculator Input

Derivative function f(x, y)

Enter the differential equation right side.

Exact solution y(x) (optional)

Used for comparison and error analysis.

Initial x value

Starting x coordinate.

Initial y value

Initial condition y(x₀).

Step size h

Smaller values usually improve accuracy.

Number of steps

Range allowed: 1 to 500.

Decimal places

Range allowed: 2 to 10.

Reset

Formula Used

Given: y′ = f(x, y), with x₀, y₀, step size h.

First slope: k₁ = f(x_i, y_i)

Midpoint location: x_mid = x_i + h / 2

Midpoint estimate: y_mid = y_i + (h / 2)k₁

Midpoint slope: k_mid = f(x_mid, y_mid)

Next value: y_i+1 = y_i + hk_mid

The midpoint method is a second-order Runge–Kutta scheme. Its global error is generally O(h²), while local truncation error is O(h³).

How to Use This Calculator

Enter the derivative expression f(x, y), such as x+y or x-y^2.
Enter the initial values x₀ and y₀.
Set the step size h and the number of steps to compute.
Optionally enter the exact solution y(x) for node-by-node error analysis.
Select the decimal precision you want in the result table.
Press Submit to view the result above the form, plus the graph and iteration table.
Use the export buttons to save the generated results as CSV or PDF.

Example Data Table

Example problem: y′ = x + y, x₀ = 0, y₀ = 1, h = 0.2, steps = 5, exact solution y = 2e^x − x − 1.

Node	x	Approximate y	Exact y	\|Error\|
0	0.000000	1.000000	1.000000	0.000000
1	0.200000	1.240000	1.242806	0.002806
2	0.400000	1.576800	1.583649	0.006849
3	0.600000	2.031696	2.044238	0.012542
4	0.800000	2.630669	2.651082	0.020413
5	1.000000	3.405416	3.436564	0.031147

Frequently Asked Questions

1) What does the midpoint method calculate?

It approximates numerical solutions of first-order ordinary differential equations. The method uses a trial midpoint slope before advancing to the next y value.

2) Why is it better than basic Euler’s method?

It usually gives better accuracy because the update uses a midpoint slope instead of only the starting slope. That improves the overall order of accuracy.

3) What expressions can I enter?

You can enter formulas using x, y, numbers, parentheses, and supported functions like sin, cos, sqrt, log, exp, abs, pi, and e.

4) Do I need an exact solution?

No. The exact solution field is optional. It only adds comparison values and error metrics such as final error, MAE, RMSE, and maximum absolute error.

5) What step size should I choose?

Smaller step sizes often improve accuracy, but they also increase the number of calculations. Try several h values and compare stability and error behavior.

6) Why does my expression fail?

Most issues come from unsupported syntax or missing multiplication symbols. Write 2*x instead of 2x, and check all parentheses carefully.

7) What does the graph show?

The chart plots midpoint approximations against x values. If an exact solution is provided, it also shows exact values and absolute error across nodes.

8) Can I export the results?

Yes. The generated iteration table can be downloaded as CSV or PDF after calculation. This is useful for reports, assignments, and later review.