Introduction to Differential Equations

By the end of this lesson, you will be able to identify and classify ordinary differential equations, verify solutions, find general and particular solutions, and understand the concept of initial value problems.

The Big Idea

A differential equation (ODE) is an equation relating a function to its derivatives. Unlike algebraic equations (solving for a number), differential equations are solved for a function. They model everything from population dynamics to electrical circuits to the motion of planets.

Classification

Order: the order of the highest derivative.

• $y' = 3y$: first-order
• $y'' + 4y = 0$: second-order

Linear vs. nonlinear: linear if $y$ and all its derivatives appear linearly (no products, powers, or compositions of the dependent variable).

• $y' + xy = \sin x$: linear
• $(y')^2 = y$: nonlinear

Verifying a Solution

Substitute the proposed solution into the ODE and confirm both sides are equal.

Example 1 — Verification

Verify $y = Ce^{3x}$ solves $y' = 3y$.

$y' = 3Ce^{3x}$ and $3y = 3Ce^{3x}$ → equal for any $C$. ✓ $y = Ce^{3x}$ is the general solution.

Example 2 — Initial Value Problem

Solve $y' = 3y$ with $y(0) = 5$.

General solution: $y = Ce^{3x}$

Apply $y(0) = 5$: $C \cdot 1 = 5 \implies C = 5$

Particular solution: $y = 5e^{3x}$

General Solution and Arbitrary Constants

An $n$th-order ODE typically has a general solution with $n$ arbitrary constants.

Specifying $n$ initial conditions (values of $y, y', \ldots, y^{(n-1)}$ at a point) determines a unique particular solution.

Example 3 — Second-Order

Verify $y = C_1\sin x + C_2\cos x$ solves $y'' + y = 0$.

$y' = C_1\cos x - C_2\sin x$

$y'' = -C_1\sin x - C_2\cos x$

$y'' + y = (-C_1\sin x - C_2\cos x) + (C_1\sin x + C_2\cos x) = 0$ ✓

Types of Solutions

Explicit solution: $y = f(x)$ — $y$ expressed as a function of $x$.

Implicit solution: $F(x, y) = 0$ — not solved for $y$.