Differential Equation Solver

A differential equation is an equation relating a function to its derivatives, describing how quantities change. General Form: F(x, y, y', y'', ..., y⁽ⁿ⁾) = 0 Where: • y = unknown function • y', y'', y⁽ⁿ⁾ = derivatives • x = independent variable Classification: 1. By Type: • ODE (Ordinary): One independent variable • PDE (Partial): Multiple independent variables 2. By Order: • First-order: Highest derivative is y' • Second-order: Highest derivative is y'' • nth-order: Highest derivative is y⁽ⁿ⁾ 3. By Linearity: • Linear: y and derivatives appear linearly • Nonlinear: Products/powers of y or derivatives Common Solution Methods: • Separation of Variables • Integrating Factors • Characteristic Equations • Variation of Parameters • Laplace Transforms Applications: Motion, heat transfer, wave propagation, population dynamics, circuits, and modeling change in natural phenomena.