Constant-Coefficient ODE Solver (Characteristic Equation): Step-by-Step Solutions

Characteristic equation, roots (with synthetic division), general solution and IVPs, up to order 6. Every step of the solution is shown, as you would write it on paper, using exact fractions.

What it solves

Enter the coefficients $a_n,\dots ,a_1,a_0$ of $a_n{y}^{(n)}+\cdots +a_1{y}^{\prime }+a_{0}y=0$ (highest derivative first). Initial conditions at $x=0$ are optional.

How to use it

1. Enter your own numbers or functions in the boxes (an example is already filled in, or press Random example).
2. Check the equation preview, then press Solve step by step.
3. Reveal the solution one step at a time with Next step, or show everything at once.

Need the theory first? Read Section 4.2: Homogeneous Linear Equations with Constant Coefficients, which explains the method in small steps with worked examples and exercises, then come back and test yourself with the Chapter 4 quiz.

All computations run in your browser; nothing is sent to a server.