Differential Equations

1. Introduction to Differential Equations

A differential equation is an equation that relates a function to one or more of its derivatives. While algebraic equations involve unknown numbers, differential equations involve unknown functions. They are the primary mathematical tool for modeling systems that change over time or space.

What Is a Differential Equation?

At its simplest, a differential equation is any equation that contains derivatives. For example:

This tells us that the rate of change of y with respect to x is 3x². Solving it means finding a function y(x) whose derivative equals 3x². In this case, y = x³ + C, where C is an arbitrary constant.

A more complex example:

Here we must find a function y(x) such that its second derivative plus five times its first derivative plus six times the function itself equals zero.

Classification of Differential Equations

ODE vs. PDE

• Ordinary Differential Equation (ODE): Contains derivatives with respect to a single independent variable. Example: dy/dx + y = eˣ.
• Partial Differential Equation (PDE): Contains partial derivatives with respect to two or more independent variables. Example: ∂u/∂t = k · ∂²u/∂x². This is the heat equation.

Order

The order of a differential equation is the highest derivative that appears in the equation.

• First order: dy/dx + y = x (highest derivative is dy/dx)
• Second order: d²y/dx² + 3 dy/dx + 2y = 0 (highest derivative is d²y/dx²)
• Third order: y‴ + y″ − y′ + y = sin(x)

Linearity

A differential equation is linear if the unknown function y and all its derivatives appear only to the first power and are not multiplied together. Otherwise it is nonlinear.

• Linear: y″ + 3y′ + 2y = eˣ
• Nonlinear: y · y′ + y² = x (the product y · y′ and the term y² make it nonlinear)
• Nonlinear: (y′)² + y = 0 (the squared derivative makes it nonlinear)

Homogeneous vs. Non-Homogeneous

A linear ODE is homogeneous if the right-hand side is zero, and non-homogeneous if it is not:

Solutions of Differential Equations

A solution to a differential equation is a function that satisfies the equation when substituted in. There are several types:

• General solution: Contains arbitrary constants and represents the entire family of solutions. For an nth-order ODE, the general solution typically contains n arbitrary constants.
• Particular solution: A specific solution obtained by assigning values to the arbitrary constants, usually via initial conditions or boundary conditions.
• Singular solution: A solution that cannot be obtained from the general solution for any value of the constants.

2. Separable Equations

A first-order ODE is separable if it can be written so that all terms involving y are on one side and all terms involving x are on the other. Separable equations are among the most common and easiest to solve.

Standard Form

A separable equation has the form:

We separate the variables:

Then integrate both sides:

3. First-Order Linear Equations

A first-order linear ODE has the standard form:

This equation is linear in y (no y², y·y′, etc.). Even when it is not separable, there is a systematic method using an integrating factor.

The Integrating Factor Method

The key idea is to multiply both sides by a carefully chosen function μ(x) — the integrating factor — that makes the left side collapse into the derivative of a product.

Step 1: Write the equation in standard form: dy/dx + P(x)·y = Q(x).

Step 2: Compute the integrating factor:

Step 3: Multiply both sides by μ(x). The left side becomes d/dx[μ(x)·y]:

Step 4: Integrate both sides:

Step 5: Solve for y:

4. Exact Equations

An equation of the form:

is called exact if there exists a function F(x, y) such that:

In that case, the solution is simply:

Test for Exactness

By Clairaut's theorem, if F has continuous second partial derivatives, then ∂²F/∂y∂x = ∂²F/∂x∂y. This means:

Solving an Exact Equation

Step 1: Verify ∂M/∂y = ∂N/∂x.

Step 2: Integrate M with respect to x (holding y constant): F = ∫ M dx + g(y), where g(y) is an unknown function of y.

Step 3: Differentiate F with respect to y and set it equal to N: ∂F/∂y = ∂/∂y[∫ M dx] + g′(y) = N. Solve for g′(y) and integrate to find g(y).

Step 4: Write the solution F(x, y) = C.

5. Second-Order Linear Equations

A second-order linear ODE with constant coefficients has the general form:

where a, b, c are real constants and a ≠ 0. These equations arise naturally in physics—especially in the study of vibrations, oscillations, and electrical circuits.

The Characteristic Equation

We guess a solution of the form y = eʳˣ. Substituting into ay″ + by′ + cy = 0 gives:

Since eʳˣ ≠ 0, we need:

The nature of the roots r₁ and r₂ determines the form of the general solution:

Case 1: Two Distinct Real Roots (b² − 4ac > 0)

Case 2: Repeated Real Root (b² − 4ac = 0)

r₁ = r₂ = r = −b/(2a). A single exponential is not enough; we need a second linearly independent solution:

Case 3: Complex Conjugate Roots (b² − 4ac < 0)

r = α ± βi, where α = −b/(2a) and β = √(4ac − b²)/(2a). Using Euler's formula:

This represents oscillatory behavior with exponential growth or decay (depending on the sign of α).

6. Non-Homogeneous Equations

A non-homogeneous second-order linear ODE has the form:

where f(x) ≠ 0 is the forcing function (or input, or driving term).

Structure of the Solution

The general solution is the sum of two parts:

• y_h — the general solution to the associated homogeneous equation (ay″ + by′ + cy = 0), called the complementary solution.
• y_p — any single particular solution to the non-homogeneous equation.

Method of Undetermined Coefficients

This method works when f(x) is a polynomial, exponential, sine, cosine, or a combination of these. We guess the form of y_p based on f(x), plug it into the equation, and solve for the unknown coefficients.

Variation of Parameters

This method works for any forcing function f(x), not just the special types above. Given a homogeneous solution y_h = C₁y₁ + C₂y₂, we seek a particular solution of the form:

where u₁ and u₂ are unknown functions determined by:

and W is the Wronskian:

7. Laplace Transforms

The Laplace transform is a powerful integral transform that converts a differential equation in the time domain into an algebraic equation in the frequency domain. After solving the algebraic equation, we apply the inverse Laplace transform to obtain the solution in the original variable.

Definition

The Laplace transform of a function f(t), defined for t ≥ 0, is:

provided the integral converges.

Common Laplace Transforms

Key Properties

• Linearity: ℒ{af + bg} = aF(s) + bG(s).
• Derivative property: ℒ{f′(t)} = sF(s) − f(0). More generally: ℒ{f⁽ⁿ⁾(t)} = sⁿF(s) − s^(n−1)f(0) − s^(n−2)f′(0) − … − f⁽ⁿ⁻¹⁾(0).
• First shifting theorem: ℒ{eᵃᵗf(t)} = F(s − a).
• Second shifting theorem: ℒ{u(t−a)·f(t−a)} = e⁻ᵃˢF(s), where u is the unit step function.
• Convolution theorem: ℒ{(f * g)(t)} = F(s)·G(s), where (f * g)(t) = ∫₀ᵗ f(τ)g(t−τ) dτ.

Solving IVPs with Laplace Transforms

The procedure is:

1. Take the Laplace transform of both sides of the ODE.
2. Use the derivative property to incorporate initial conditions.
3. Solve the resulting algebraic equation for Y(s) = ℒ{y(t)}.
4. Apply the inverse Laplace transform to find y(t).

8. Systems of Differential Equations

Many real-world problems involve multiple interacting quantities, each changing over time. These are modeled by systems of differential equations—a set of coupled ODEs.

General Form

A system of two first-order linear ODEs can be written as:

In matrix form:

where A is the coefficient matrix.

Solving Homogeneous Systems with Eigenvalues

For the homogeneous system x′ = Ax, we look for solutions of the form x(t) = veᵏᵗ, where v is a constant vector and λ is a scalar. Substituting gives:

So λ must be an eigenvalue of A, and v its corresponding eigenvector.

The general solution is:

Converting Higher-Order to Systems

Any nth-order ODE can be converted to a system of n first-order ODEs. For y″ + py′ + qy = f(t), let x₁ = y and x₂ = y′. Then:

This technique is essential for numerical methods and theoretical analysis.

9. Applications

Differential equations are the backbone of mathematical modeling in science and engineering. Here are some of the most important applications.

Population Growth Models

Exponential Growth

The simplest model assumes the rate of growth is proportional to the current population:

This is separable: P(t) = P₀eᵏᵗ, where P₀ is the initial population and k is the growth rate constant.

Logistic Growth

Exponential growth is unrealistic for large populations due to limited resources. The logistic equation introduces a carrying capacity K:

When P is small compared to K, growth is approximately exponential. As P approaches K, growth slows and eventually stops.

Newton's Law of Cooling

The rate of change of an object's temperature is proportional to the difference between its temperature and the ambient temperature T_a:

This is a first-order linear ODE with solution:

Spring-Mass Systems (Harmonic Oscillators)

A mass m attached to a spring with spring constant k (and damping coefficient c) satisfies:

where y is the displacement from equilibrium and F(t) is any external force.

• Undamped free vibration (c = 0, F = 0): my″ + ky = 0 → y = A cos(ω₀t − φ), where ω₀ = √(k/m) is the natural frequency.
• Damped free vibration (F = 0): Depends on whether c² − 4mk is positive (overdamped), zero (critically damped), or negative (underdamped).
• Forced vibration: Adding F(t) = F₀ cos(ωt) creates resonance phenomena if ω ≈ ω₀.

Electrical Circuits (RLC Circuits)

An RLC series circuit with resistance R, inductance L, and capacitance C, driven by a voltage E(t), satisfies:

Since I = dq/dt, this becomes a second-order ODE in the charge q:

This is mathematically identical to the spring-mass equation! The analogy is: L ↔ m, R ↔ c, 1/C ↔ k, E(t) ↔ F(t).

10. Introduction to Partial Differential Equations

Partial differential equations (PDEs) involve functions of several variables and their partial derivatives. PDEs are essential for describing phenomena that depend on multiple independent variables, such as position and time.

The Heat Equation

The heat equation models how temperature u(x, t) evolves in a rod over time:

where k > 0 is the thermal diffusivity. This is a parabolic PDE.

To solve the heat equation, we typically need:

• Initial condition: u(x, 0) = f(x) (initial temperature distribution).
• Boundary conditions: e.g., u(0, t) = 0, u(L, t) = 0 (ends held at zero temperature).

Separation of Variables

We assume u(x, t) = X(x)·T(t). Substituting into the PDE:

Since the left side depends only on t and the right only on x, both must equal a constant −λ. This gives two ODEs:

With boundary conditions u(0,t) = u(L,t) = 0, we need X(0) = 0 and X(L) = 0. The eigenvalues and eigenfunctions are:

The corresponding time solutions are:

The general solution is a Fourier sine series:

where the coefficients Bₙ are determined by the initial condition via Bₙ = (2/L) ∫₀ᴸ f(x) sin(nπx/L) dx.

The Wave Equation

The wave equation models vibrations (strings, membranes, sound):

where c is the wave speed. This is a hyperbolic PDE.

For a string of length L fixed at both ends (u(0,t) = u(L,t) = 0), with initial displacement u(x,0) = f(x) and initial velocity uₜ(x,0) = g(x), separation of variables gives:

where:

Laplace's Equation

Laplace's equation models steady-state (time-independent) phenomena such as electrostatics, steady-state heat distribution, and incompressible fluid flow:

This is an elliptic PDE. Solutions are called harmonic functions and have many remarkable properties:

• The value at any point equals the average of the values on any circle centered at that point.
• The maximum and minimum values occur on the boundary, not in the interior (Maximum Principle).
• Solutions are infinitely differentiable (smooth).

Classification of Second-Order PDEs

A general second-order PDE in two variables can be written as:

The classification depends on the discriminant B² − AC:

• B² − AC > 0 → Hyperbolic (wave-like behavior, e.g., wave equation)
• B² − AC = 0 → Parabolic (diffusion-like behavior, e.g., heat equation)
• B² − AC < 0 → Elliptic (steady-state behavior, e.g., Laplace equation)