Differential Equation Solver

Analytic solutions of constant-coefficient linear ODEs of first and second order, and RK4 numerical solutions of y′ = f(x, y).

Your result will appear here

Fill in the fields and press Solve, or pick one of the examples.

About the Differential Equation Solver

Solves a deliberately limited set of ordinary differential equations, and is explicit about which answers are analytic and which are numerical.

Analytic: first-order linear equations with constant coefficients, $y' + ay = b$ , and second-order linear homogeneous equations with constant coefficients, $ay'' + by' + cy = 0$ (all three cases of the characteristic equation). Numerical: any initial value problem y' = f(x, y), integrated with the classical fourth-order Runge–Kutta method.

Limitations

No general symbolic ODE solver: separable, exact, non-homogeneous second-order and variable-coefficient equations are not solved analytically.
Coefficients in analytic solutions are shown as decimals.
RK4 results are approximations; the error estimate comes from repeating the computation with half the step size.

How to use it

Choose the type of equation.
Enter the coefficients (or the right-hand side f(x, y)) and, optionally, initial conditions.
Press Solve. With initial conditions the solution is also plotted.

Example

y'' + y = 0, y(0) = 0, y'(0) = 1

→ y = sin(x)

Formulas used

y' + ay = b \;\Rightarrow\; y = \frac{b}{a} + Ce^{-ax}

First-order linear, constant coefficients (a ≠ 0).

ar^{2} + br + c = 0

Characteristic equation of ay'' + by' + cy = 0; its roots determine the form of the solution.

Integral CalculatorIndefinite integrals (antiderivatives) found symbolically and verified by differentiation before they are shown.Numerical Integration CalculatorApproximate integrals with the trapezoidal, midpoint and Simpson’s rules and compare them with an adaptive reference value.Graphing CalculatorPlot and compare functions, implicit equations and inequalities with zoom, pan, coordinate tracking and PNG export.

About the Differential Equation Solver

Limitations

How to use it

Formulas used

Related tools

About the Differential Equation Solver

Limitations

How to use it

Formulas used