Introduction and Classification

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.

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.

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.

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.

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)

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

• Confusing order with degree. The order of an ODE is the highest derivative, not the highest power of $y$ or $y'$.
• Misidentifying linearity. A term like $yy'$ or $(y')^2$ makes an ODE nonlinear, even if each derivative appears only once.
• Mixing up IVP and BVP. An IVP specifies conditions at a single point; a BVP specifies conditions at two or more distinct points.
• Assuming superposition for nonlinear equations. The superposition principle applies only to linear homogeneous ODEs.

• Engineering: Electrical circuit analysis (RLC circuits) and control systems rely on linear ODEs with constant coefficients.
• Biology: Epidemiological models (SIR equations) and population dynamics use nonlinear systems of ODEs.
• Physics: Newtonian mechanics, quantum mechanics (Schrödinger equation), and general relativity (Einstein field equations) are formulated as differential equations.
• Economics: Black-Scholes equation for option pricing is a PDE; macroeconomic growth models use ODE systems.