Differential Equations

Differential Equation

An equation that relates one or more functions and their derivatives.

Order of a Differential Equation

The order of the highest derivative present in the equation.

Ordinary Differential Equation (ODE)

A differential equation containing derivatives of a function of only one independent variable.

Partial Differential Equation (PDE)

A differential equation containing partial derivatives of a function of two or more independent variables.

General Solution

The solution of a differential equation that contains an arbitrary constant (or constants) equal to the order of the equation.

Particular Solution

A solution obtained from the general solution by assigning specific values to the arbitrary constants, usually determined by initial or boundary conditions.

Initial Conditions

Values of the function and its derivatives at a single point, used to determine the particular solution of an ODE.

Boundary Conditions

Values of the function or its derivatives at different points, used to determine the particular solution, especially for PDEs.

Integrating Factor for First-Order Linear ODE

For equations of the form dy/dx + P(x)y = Q(x)

I(x) = e^{\int P(x) dx}

Characteristic Equation for Second-Order Linear Homogeneous ODE with Constant Coefficients

For equations of the form ay'' + by' + cy = 0

ar² + br + c = 0

General Solution (Real Distinct Roots)

When the characteristic equation has two distinct real roots r1 and r2.

y = C₁ e^{r_1 x} + C₂ e^{r_2 x}

General Solution (Real Repeated Roots)

When the characteristic equation has one real repeated root r.

y = (C₁ + C₂ x) e^rx

General Solution (Complex Roots)

When the characteristic equation has complex roots \α \± i\β.

y = e^{\α x} (C₁ \cos(\β x) + C₂ \sin(\β x))