Introduction and Classification

1.1 Basic Definitions

A differential equation (DE) is an equation involving an unknown function and its derivatives. An ordinary differential equation (ODE) involves a function of one variable and its ordinary Derivatives. A partial differential equation (PDE) involves a function of several variables and Its partial derivatives.

1.2 Classification of ODEs

An ODE is:

• Ordinary vs. partial: depends on whether partial derivatives appear.
• Order: the highest derivative that appears.
• Linear vs. nonlinear: linear if the unknown function and its derivatives appear linearly.
• Homogeneous vs. nonhomogeneous: for linear ODEs, homogeneous if the forcing term is zero.

1.3 Initial and Boundary Value Problems

An initial value problem (IVP) specifies the value of the function (and possibly its Derivatives) at a single point. A boundary value problem (BVP) specifies conditions at two or More points.

1.4 Examples from Physics and Biology

Differential equations arise throughout the natural sciences. A few canonical examples:

1. Newton”s law of cooling. The temperature $T(t)$ of a body in a medium at temperature $T_m$ satisfies $\frac{dT}{dt} = -k(T - T_m)$A first-order linear ODE.

2. Harmonic oscillator. A mass on a spring with damping obeys $m\frac{d^2 x}{dt^2} + c\frac{dx}{dt} + kx = F(t)$A second-order linear ODE.

3. Logistic population growth. $\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$A first-order nonlinear (Bernoulli) ODE.

4. Lotka-Volterra predator-prey model. $\frac{dx}{dt} = x(\alpha - \beta y)$, $\frac{dy}{dt} = y(-\gamma + \delta x)$A coupled nonlinear system.

5. RC circuit. The charge $q(t)$ on a capacitor satisfies $R\frac{dq}{dt} + \frac{q}{C} = V(t)$ a first-order linear ODE.

6. Heat equation. The temperature $u(x, t)$ in a rod satisfies $u_t = \alpha^2 u_{xx}$A second-order linear PDE.

7. Wave equation. The displacement $u(x, t)$ of a string satisfies $u_{tt} = c^2 u_{xx}$A second-order linear PDE.

8. Laplace’s equation. The steady-state temperature satisfies $u_{xx} + u_{yy} = 0$A second-order linear PDE.

1.5 Classification Tree

Differential Equations
├── ODE (one independent variable)
│   ├── By order
│   │   ├── First-order: y' = f(x, y)
│   │   ├── Second-order: y'' = f(x, y, y')
│   │   └── n-th order: y^(n) = f(x, y, ..., y^(n-1))
│   ├── By linearity
│   │   ├── Linear: a_n(x)y^(n) + ... + a_0(x)y = g(x)
│   │   │   ├── Homogeneous (g = 0)
│   │   │   └── Nonhomogeneous (g ≠ 0)
│   │   └── Nonlinear (y or derivatives appear nonlinearly)
│   └── By coefficients
│       ├── Constant coefficient
│       └── Variable coefficient
└── PDE (multiple independent variables)
    ├── Elliptic: B² - 4AC < 0  (e.g., Laplace)
    ├── Parabolic: B² - 4AC = 0 (e.g., Heat)
    └── Hyperbolic: B² - 4AC > 0 (e.g., Wave)

1.6 Worked Example: Classifying ODEs

Problem. Classify each equation by order, linearity, and homogeneity (if linear).

(a) $y'' + 3y' + 2y = \sin x$

(b) $(y')^2 + y = 0$

(c) $x^2 y'' + xy' + (x^2 - 1)y = 0$

(d) $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$