MathCalcLab

Differential Equations Calculator

Solve ordinary differential equations (ODEs) with step-by-step solutions. Supports separable equations, linear ODEs, and initial value problems with graphical visualization.

Differential Equation Setup

Examples: y, x, x*y, x/y

Slope Field Visualization

Each small line segment shows the slope dy/dx at that point. The orange dot shows your initial condition.

Theory & Formula

Differential Equations

A differential equation relates a function to its derivatives. First-order ODEs have the form dy/dx = f(x,y). The solution is a function y(x) that satisfies this relationship.

Separable Equations

A separable equation can be written as a product of functions of x and y. We can separate variables and integrate both sides:

\(\frac{dy}{dx} = g(x)h(y) \implies \frac{dy}{h(y)} = g(x)\,dx\)

Example: Solve dy/dx = y

\(\frac{dy}{dx} = y\)\(\frac{dy}{y} = dx\)\(\int \frac{dy}{y} = \int dx\)\(\ln|y| = x + C_1\)\(y = Ce^x\)

Initial Value Problems

An initial value problem (IVP) specifies both a differential equation and an initial condition y(x₀) = y₀. This determines a unique particular solution.