Differential Equations

Ordinary Differential Equation (ODE)

An equation involving an unknown function of one independent variable and its derivatives.

Partial Differential Equation (PDE)

An equation involving an unknown function of two or more independent variables and its partial derivatives.

Order of a Differential Equation

The order of the highest derivative present in the equation.

Degree of a Differential Equation

The highest power of the highest order derivative, after the equation has been cleared of radicals and fractional exponents of derivatives.

General Solution

The solution containing as many arbitrary constants as the order of the differential equation.

Particular Solution

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

Complementary Function (CF)

The general solution of the associated homogeneous differential equation.

Particular Integral (PI)

Any particular solution of the non-homogeneous differential equation.

Integrating Factor for Linear First-Order ODE

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

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

General Solution of Linear First-Order ODE

Where I(x) is the integrating factor.

y \· I(x) = \int Q(x) \· I(x) dx + C

Characteristic Equation for Second-Order Linear ODE with Constant Coefficients

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

am² + bm + c = 0

Solution for Distinct Real Roots (m1, m2)

When the characteristic equation has two distinct real roots.

y(x) = C₁ e^{m_1 x} + C₂ e^{m_2 x}

Solution for Repeated Real Root (m)

When the characteristic equation has one real root repeated.

y(x) = (C₁ + C₂ x) e^{m x}

Solution for Complex Conjugate Roots (α ± iβ)

When the characteristic equation has complex conjugate roots.

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