Differential Equation Solver

An ordinary differential equation is an equation that relates a function y(x) with one or more of its derivatives (y', y'', etc.) with respect to a single independent variable x. ODEs appear throughout physics, engineering, biology, and economics to model rates of change and dynamic systems.

A general solution contains one or more arbitrary constants (C₁, C₂, etc.) and represents the entire family of solutions to the ODE. A particular solution is obtained by applying initial conditions — known values of y and/or y' at a specific x — to determine the exact constants.

This calculator handles first-order linear, first-order separable, first-order exponential growth/decay, second-order homogeneous, and second-order inhomogeneous (constant-forcing) ordinary differential equations. For more complex equations involving partial derivatives or non-constant coefficients, specialized symbolic solvers are recommended.

For a second-order linear ODE with characteristic equation ar² + br + c = 0, the discriminant Δ = b² - 4ac determines the nature of the roots: if Δ > 0, two distinct real roots (overdamped); if Δ = 0, one repeated real root (critically damped); if Δ < 0, two complex conjugate roots (underdamped oscillation).

For a second-order ODE, you need two initial conditions: y(x₀) = y₀ (the value of the function at x₀) and y'(x₀) = y'₀ (the value of the derivative at x₀). Enter x₀, y₀, and y'₀ in the Initial Conditions section and select 'Yes — find particular solution'.

A separable ODE is one that can be written in the form dy/dx = f(x) · g(y), where the variables x and y can be separated to opposite sides of the equation. The solution method involves integrating both sides independently. A classic example is dy/dx = ky, which gives exponential growth or decay.

Stability refers to whether solutions to an ODE grow, decay, or oscillate over time. For second-order systems, if both roots of the characteristic equation have negative real parts, the system is stable (solutions decay to zero). Positive real parts indicate instability. Pure imaginary roots produce sustained oscillations.

This tool is designed for constant-coefficient linear ODEs, which cover a wide range of practical problems. Nonlinear ODEs (e.g., y'' + sin(y) = 0) or equations with variable coefficients generally require numerical methods like Euler's method, Runge-Kutta (RK4), or symbolic solvers like Wolfram Alpha or MATLAB's ode45.