Differential Equation Calculator
A Differential Equation Calculator is an online tool that solves ordinary and partial differential equations, offering analytical and numerical solutions with step-by-step explanations, simplifying complex mathematical problems for students, engineers, and researchers.
diff(y,x) for derivative. Example: diff(y,x) - y = 0Button Pad
Examples
Equation Preview:
Result
What is Differential Equation Calculator?
A Differential Equation Calculator is a tool that helps solve equations involving derivatives — relationships that describe how quantities change. It supports ordinary differential equations (ODEs) and often partial differential equations (PDEs), returning either symbolic (analytical) solutions when possible or numeric approximations otherwise. Typical problems include initial-value problems, boundary-value problems, linear and nonlinear equations, separable equations, and higher-order linear ODEs.
About the Differential Equation Calculator
The calculator applies standard methods: separation of variables, integrating factors for linear first-order ODEs, characteristic equations for constant-coefficient linear ODEs, and numerical solvers (e.g., Runge–Kutta) when symbolic integration fails. It displays intermediate steps to teach the method and provides options to plot solutions, compare numerical vs. analytic results, and export results.
For constant-coefficient second-order linear ODEs:
solve the characteristic equation to form the general solution using the roots \(r\).
How to Use this Differential Equation Calculator
- Choose equation type (first-order, second-order, system, or PDE) and input the equation using \(x, y, t,\) or other variables.
- If applicable, provide initial or boundary conditions (e.g., \(y(0)=y_0\)).
- Select preferred method: symbolic (attempt analytic solution) or numeric (choose solver and step-size).
- Click Calculate — the tool shows step-by-step transformations, applied formulas (shown above), and the final solution or numeric output.
- Use plotting options to visualize \(y(x)\) or solution surfaces, and export results if needed.
Example (first-order linear): Solve \(\frac{dy}{dx}+2y=\sin x\). Integrating factor \(\mu=e^{\int 2\,dx}=e^{2x}\), hence \[ y=\frac{1}{e^{2x}}\left(\int e^{2x}\sin x\,dx + C\right). \]
This calculator is useful for students and professionals tackling dynamics, circuits, population models, heat and wave problems, and any application modeled by differential equations.