Diff Eq Solver

A differential equation (diff eq) is an equation relating a function to its derivatives, expressing how quantities change. Basic Form: Equation involving y, y', y'', etc. Types: • ODE: Ordinary (one variable) • PDE: Partial (multiple variables) Order: • First: Contains y' • Second: Contains y'' • Higher: Contains y⁽ⁿ⁾ Linearity: • Linear: y and derivatives appear linearly • Nonlinear: Products or powers present Common Forms: 1. First-Order: dy/dx = f(x,y) 2. Second-Order Linear: ay'' + by' + cy = g(x) 3. Separable: dy/dx = g(x)h(y) Solution Types: • General Solution: Contains arbitrary constants • Particular Solution: Specific values for constants • Initial Value Problem: Given y(x₀) = y₀ Key Methods: • Separation of Variables • Integrating Factors • Characteristic Equations • Laplace Transforms Applications: Physics (motion, waves, heat), biology (population), engineering (circuits, control), and economics (growth models).