First Order Differential Equation Calculator

Calculator Inputs

Equation Type

Numerical Method

f(x,y)

P(x) for Linear Form

Q(x) for Linear Form

Exact y(x), Optional

Initial x0

Initial y0

Target x

Step Size

Maximum Steps

Allowed Functions

sin, cos, tan, exp, log, sqrt, abs, pow, min, max, pi, e

Example Data Table

Case	Equation	y0	Target x	h	Method	Exact y(x)
Growth Check	dy/dx = x + y	1	1	0.1	RK4	2*exp(x)-x-1
Linear Check	y' + 1y = x	1	1	0.1	Compare All	x-1+2*exp(-x)
Decay Check	dy/dx = -0.5*y	10	2	0.2	Heun	10exp(-0.5x)

Formula Used

For a direct first order model, use y' = f(x,y) with y(x0) = y0.

For a linear model, y' + P(x)y = Q(x) becomes y' = Q(x) - P(x)y.

Euler uses y(n+1) = y(n) + h f(x(n), y(n)).

Heun uses the average of the first slope and the predicted end slope.

Midpoint uses a half-step trial value to estimate the full-step slope.

RK4 uses k1, k2, k3, and k4. Then y(n+1) = y(n) + h(k1 + 2k2 + 2k3 + k4)/6.

How to Use This Calculator

Select direct form or linear form.
Enter the derivative expression or P(x) and Q(x).
Add x0, y0, target x, and step size.
Choose one numerical method or compare all methods.
Add an exact expression only when it is known.
Press Calculate to show results above the form.
Use CSV or PDF buttons to export the same calculation.

Why This Calculator Matters

A first order differential equation connects a changing value with its rate of change. It appears in motion, cooling, growth, decay, circuits, mixing, and finance models. A calculator helps because hand steps can become repetitive. It also reduces small arithmetic errors.

Numerical Insight

This tool focuses on initial value problems. You enter x0, y0, a target x value, and a step size. The model may be written directly as dy/dx = f(x,y). You may also use the linear form y' + P(x)y = Q(x). The page converts that form into a derivative rule.

Method Comparison

Euler is simple and fast. It uses the current slope only. The midpoint method checks the slope halfway through the step. Heun uses an average of the starting and predicted slopes. RK4 blends four slopes. It usually gives stronger accuracy for smooth equations.

Advanced Options

You can choose one method or compare all methods. You can move forward or backward along the x axis. The final step is adjusted so the table ends exactly at the target. An optional exact expression can estimate error at every step. This is useful in lessons and verification.

Practical Use

Start with a small step size when accuracy matters. Then run the same problem with a smaller step. If the answer changes very little, the result is more dependable. If the answer changes a lot, the equation may need smaller steps, a shorter interval, or expert review.

Readable Output

The result table shows each step, x value, y value, slope data, exact value, and absolute error when available. CSV export supports spreadsheets. PDF export creates a compact report for records. The example table gives quick test cases before entering custom equations.

Common Modeling Checks

Before trusting any output, confirm the equation matches the real situation. Check signs, units, and starting conditions. A negative sign can reverse growth into decay. A large step can hide rapid changes. Discontinuous formulas may fail near jumps or undefined points. Compare methods when possible. Close agreement is a helpful signal, not a proof. For formal work, combine the table with theory, graphs, and boundary knowledge. This calculator supports exploration, homework checks, and planning notes. Always record assumptions beside exported results for later review.

FAQs

What is a first order differential equation?

It is an equation involving a function and its first derivative. It describes how a value changes with respect to one independent variable, often written as dy/dx = f(x,y).

Which method should I choose?

RK4 is often a strong default for smooth equations. Euler is best for learning basic steps. Heun and midpoint give useful comparisons without much extra complexity.

Can I solve linear equations here?

Yes. Select the linear option. Enter P(x) and Q(x). The calculator converts y' + P(x)y = Q(x) into y' = Q(x) - P(x)y.

What functions are supported?

You can use common functions such as sin, cos, tan, exp, log, sqrt, abs, pow, min, and max. Constants pi and e are also supported.

Why is an exact expression optional?

Many differential equations have no simple closed form. When an exact solution is known, entering it lets the table show absolute error at each step.

Can the calculator work backward?

Yes. Enter a target x smaller than x0. The calculator automatically uses negative steps and adjusts the last step to finish at the target.

Why does step size matter?

Smaller steps usually improve accuracy, but they create longer tables. Large steps may skip important behavior or increase error, especially for rapidly changing equations.

What do the CSV and PDF buttons do?

The CSV button exports the full step table for spreadsheet work. The PDF button creates a compact report with final answers and selected table rows.