Differential equation solver

Types of differential equations

First order linear: y' + P(x)y = Q(x). Solution uses integrating factor e^(∫P dx).

Separable: dy/dx = f(x)g(y). Separate variables and integrate both sides.

Second order constant: ay'' + by' + cy = 0. Use characteristic equation ar² + br + c = 0.

Exact: M(x,y)dx + N(x,y)dy = 0 where ∂M/∂y = ∂N/∂x.

Notation

Use y' for dy/dx, y'' for d²y/dx². Or use Leibniz notation: dy/dx, d2y/dx2.

Example: y' + 2y = 0

This is first-order linear. Separating: dy/y = -2dx. Integrating: ln|y| = -2x + C. Solution: y = Ce^(-2x).