Differential Equation Calculator
Differential Equation Calculator works out General solution and Particular solution from How to solve it, Differential equation and Independent variable. The default figures on this page give general solution of y = 2 + c * exp(-2 * x); change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
5 steps
- Read as 0 y'' + 1 y' + 2 y = 4.
- Written in standard form: y' + (2) y = 4
- Integrating factor μ = exp(∫2 dx) = exp(2x). Multiplying through and integrating gives y = 4/2 + C exp(-2x).
- General solution: y = 2 + c * exp(-2 * x)
- Check: substituting this back into the equation leaves a residual of essentially zero.
Its 5 worked examples are re-run on every release, last on 15 September 2026. What changed
Sources and references
3 sources
What the Differential Equation Calculator calculates
Solves linear ordinary differential equations exactly — first order by the integrating factor and second order with constant coefficients by the characteristic equation — applies initial conditions, and falls back to a fourth-order Runge–Kutta initial-value solver, checking every symbolic solution by substituting it back.
How to use this Differential Equation Calculator
Enter How to solve it, Differential equation, Independent variable and Initial point x₀. General solution updates as you type, and “Use a worked example” loads checked figures you can adjust. Change one figure at a time to see what drives the result.
How the result is worked out
Open “How this is worked out” beside the result for the step-by-step calculation behind General solution, Particular solution and Kind of equation.
The method follows OpenStax, Calculus Volume 2, chapter 4: Introduction to Differential Equations (separable equations and first-order linear equations), OpenStax, Calculus Volume 3, chapter 7: Second-Order Differential Equations (the characteristic equation and its three cases) and W. H. Press et al., Numerical Recipes, 3rd edition, section 17.1: the classical fourth-order Runge–Kutta method.
Decay: y' + 2y = 0 with y(0) = 1 example
With How to solve it: symbolic, Differential equation: y' + 2y = 0, Independent variable: x, Initial point x₀: 0, the calculator returns y at the requested point: 0.135335283, Order: 1. This example is checked against an independent source in the test suite.
What you can compare
Change one input at a time to see how it affects General solution, Particular solution and Kind of equation. The inputs you can vary are How to solve it, Differential equation, Independent variable, Initial point x₀, Initial value y(x₀) and Initial slope y'(x₀).
Results are general information, not financial, tax, legal, medical or engineering advice. Check any decision that matters with a qualified professional.
See something wrong? Report a problem with this calculator
Frequently asked questions
How to do a differential equation in a calculator?
Solves linear ordinary differential equations exactly — first order by the integrating factor and second order with constant coefficients by the characteristic equation — applies initial conditions. Open “How this is worked out” beside the result to see each step for your own figures.
What is the best calculator for differential equations?
Solves linear ordinary differential equations exactly — first order by the integrating factor and second order with constant coefficients by the characteristic equation — applies initial conditions. On this page it is computed from How to solve it, Differential equation and Independent variable and shown as General solution, Particular solution and Kind of equation.
Do I need Calc 3 for differential equations?
Solves linear ordinary differential equations exactly — first order by the integrating factor and second order with constant coefficients by the characteristic equation — applies initial conditions. On the figures this page opens with — initial point x₀ 0, initial value y(x₀) 1 and evaluate the solution at 1 — the general solution is y = 2 + c * exp(-2 * x). Put your own figures in their place and the working updates beside the result.
What information do I need for the Differential Equation Calculator?
You need How to solve it, Differential equation, Independent variable, Initial point x₀, Initial value y(x₀), Initial slope y'(x₀), Evaluate the solution at and Numeric mode: f(x, y) in y' = f(x, y). The page opens with default figures that give general solution of y = 2 + c * exp(-2 * x), so you can see the scale before entering your own.
What does this Differential Equation Calculator show?
It shows General solution, Particular solution, Kind of equation, Order, y at the requested point, Constants fixed by the initial conditions, How it was solved and Checked by substituting back, with the working and any warnings for your entries.
Can I check how the Differential Equation Calculator result was calculated?
Yes. The “How this is worked out” section shows the calculation steps, and the sources section identifies the references used for the method.