How to use the differential equation calculator
Start by writing the right-hand side of your equation in terms of x and y. For example, if the equation is y' = x + y, enter x + y. Then enter the starting x value and the corresponding starting y value.
Choose an end x value to set the range you want to calculate. The number of calculation steps controls how closely the numerical method follows the curve. More steps can improve the approximation, but they do not turn a numerical calculation into a symbolic solution.
Example: solve y' = x + y
Suppose the initial condition is y(0) = 1. Enter x + y, set the starting x value to 0, the starting y value to 1, and choose an end x value such as 2. The calculator then estimates the values of y along that interval.
What is a differential equation?
A differential equation describes a relationship involving a function and one or more of its derivatives. They are used to model change in areas such as physics, engineering, biology, economics, and applied mathematics.
A first-order ordinary differential equation contains a first derivative. A common numerical form is y' = f(x,y). When an initial value is known, a numerical method can estimate the solution at other x values.
How the calculation works
This page uses the fourth-order Runge-Kutta method, often abbreviated as RK4. Instead of trying to rearrange the equation into a symbolic formula, RK4 samples the slope several times during each step and combines those slope estimates to produce the next value of y.
The step size is determined by the interval and the number of steps you choose. Smaller steps generally give a closer numerical approximation when the equation behaves well over the selected interval.
General form
y' = f(x,y)
with an initial condition such as y(x₀) = y₀.
When to use a numerical solution
A numerical differential equation solver is useful when an equation is difficult to solve by hand, when you need values over a particular interval, or when you are studying how a model changes from a known starting point. For coursework, it is still worth checking the result against the method required by your class or textbook.
Understanding the result
The table gives approximate x and y values along the selected interval. These values describe points on the numerical solution. If you increase the number of steps and the values change noticeably, the problem may need a smaller step size or additional mathematical analysis.
Frequently asked questions
Can this solve every differential equation?
No. This calculator is designed for first-order ordinary differential equations that can be written as y' = f(x,y). Higher-order, partial, implicit, or equations requiring special symbolic techniques may need a different method.
Does it give an exact answer?
No. It gives a numerical approximation using RK4. The quality of the approximation depends on the equation, interval, and step size.
What does y' mean?
y' is the first derivative of y with respect to x. It describes how y changes as x changes.
What functions can I enter?
The calculator supports basic arithmetic, powers, parentheses, x and y, plus common functions including sin, cos, tan, exp, log, sqrt, and abs.
Is this a differential equation solver online, or do I need to install anything?
Everything runs right here in your browser. There is nothing to install and no account needed, so it works the same on a phone as it does on a laptop.