Differential Equations: Complete Guide

§ 01What Is a Differential Equation?

A differential equation is an equation that contains a function and one or more of its derivatives. Instead of solving for a number, you solve for an entire function.

In ordinary algebra, an equation like 3x + 1 = 7 has a number as its solution: x = 2. A differential equation works differently: its solution is a whole function, not a number. For example:

A simple first-order ODE

dy/dx = 2x

The solution is any function whose derivative equals 2x. From integration you know that is y = x² + C for any constant C. That family of parabolas, one for each value of C, is the general solution. If you are also told that y(0) = 3, that pins down C = 3, giving the particular solution y = x² + 3.

This might seem like a small step beyond integration, but differential equations become vastly richer when the derivative appears alongside y itself, when the rate of change of a quantity depends on the quantity's current value. That situation describes almost every natural process: compound interest, radioactive decay, population growth, heat transfer, electrical circuits, mechanical vibrations, and epidemics.

Why differential equations matter Newton's second law F = ma is a differential equation (force equals mass times the second derivative of position). Maxwell's equations governing electromagnetism are differential equations. The Schrödinger equation of quantum mechanics is a differential equation. Almost every law of physics is written as one.

Order and Linearity

The order of a differential equation is the order of the highest derivative it contains. An equation involving y′ (but not y″ or higher) is first-order; one involving y″ is second-order.

A differential equation is linear if y and all its derivatives appear only to the first power and are not multiplied together. Linear ODEs have solution methods that work systematically, they are the primary focus of a first course. Nonlinear ODEs are generally far harder and often have no closed-form solution.

First-order linear

y′ + P(x)y = Q(x)

Solved by integrating factor

Separable

dy/dx = f(x) · g(y)

Separate variables, integrate both sides

Second-order linear homogeneous

ay″ + by′ + cy = 0

Solved via characteristic equation

Second-order non-homogeneous

ay″ + by′ + cy = g(x)

General solution = y_h + y_p

Exponential growth / decay

dy/dt = ky

Solution: y = y₀e^kt

Laplace domain

ℒ{y″} = s²Y − sy(0) − y′(0)

Converts ODE to algebra in s

§ 02How to Work Through This Section

Differential equations build on each other. The pathway below shows the natural order, follow it from left to right for the smoothest progression.

Recommended Study Pathway

Start here Intro to ODEs → Separable Equations → Integrating Factor → First-Order Linear

Then Homogeneous Eqs. → Characteristic Eq. → Second-Order Linear

Advanced Undetermined Coefficients → Variation of Parameters → Laplace Transforms

Prerequisites: solid integration skills, especially integration by parts and partial fractions. If you're rusty, visit the Integration hub first.

§ 03All 10 Topics

Pick any topic below to get started. Each page covers the theory, several worked examples, and a 10-question quiz.

01 · First-Order Introduction to ODEs What a differential equation is, how to classify by order and linearity, general vs particular solutions, and initial value problems. Foundation 02 · First-Order Separable Equations Equations of the form dy/dx = f(x)·g(y). Separate variables, integrate both sides, apply initial conditions. Includes exponential growth and logistic models. First-Order 03 · First-Order Integrating Factor Derivation of μ(x) = e^∫P dx and why multiplying through makes the left-hand side a perfect derivative. Full worked derivation and 10 examples. First-Order 04 · First-Order First-Order Linear ODEs The standard form y′ + P(x)y = Q(x) solved completely using the integrating factor. Mixing problems, Newton's cooling, circuit equations. First-Order 05 · First-Order Homogeneous Equations Equations where dy/dx = F(y/x). The substitution v = y/x converts them to separable form. Step-by-step technique with 8 worked examples. First-Order 06 · Second-Order The Characteristic Equation Why assuming y = e^(rx) reduces ay″ + by′ + cy = 0 to a quadratic. All three root cases — real distinct, repeated, complex — and what they mean physically. Second-Order 07 · Second-Order Second-Order Linear ODEs Homogeneous equations ay″ + by′ + cy = 0 solved completely for all three discriminant cases: overdamped, critically damped, underdamped. Second-Order 08 · Second-Order Undetermined Coefficients Solving ay″ + by′ + cy = g(x) by guessing a trial particular solution based on the form of g(x). Includes the modification rule for overlapping solutions. Second-Order 09 · Second-Order Variation of Parameters A general method for non-homogeneous ODEs that works when undetermined coefficients fails. Uses the Wronskian and replaces constants with functions u₁(x), u₂(x). Advanced 10 · Advanced Laplace Transforms Converts an ODE into an algebraic equation in the s-domain. Definition, transform table, solving IVPs, inverse Laplace via partial fractions, convolution theorem. Advanced

§ 04Which Method Should I Use?

Not sure which technique applies to your equation? Answer two questions and we'll point you to the right method.

ODE Method Selector

Select the order and structure of your equation:

Order

Type / Structure

Recommended Method

§ 05The Big Ideas — A Conceptual Map

Differential equations can feel like a bag of disconnected tricks. Here is the underlying logic that ties all the methods together.

General + Particular = Complete Solution

For any linear ODE, the full solution has exactly two parts. The homogeneous solution y_h solves the equation with zero on the right-hand side, it describes the system's natural behaviour with no external driving force. The particular solution y_p is any single function that satisfies the full equation with the driving term present. Add them:

General solution structure

y = y_h + y_p

The constants in y_h (there will be as many as the order of the ODE) are determined by initial conditions. This structure is not a trick, it follows directly from the linearity of the operator: if L[y_h] = 0 and L[y_p] = g, then L[y_h + y_p] = 0 + g = g.

Why Exponentials Dominate

The function e^rx is special because differentiating it just multiplies it by r. So when you substitute y = e^rx into a constant-coefficient linear ODE, every term produces a factor of e^rx which cancels, leaving a purely algebraic equation in r, the characteristic equation. This is why exponentials appear in the solutions of virtually every linear ODE with constant coefficients.

Complex Roots Give Oscillations

When the characteristic equation has complex roots α ± βi, the exponential solutions are e^(α+βi)x and e^(α−βi)x. By Euler's formula these combine into real-valued functions e^αxcos(βx) and e^αxsin(βx), oscillations that grow if α > 0, decay if α < 0, and stay constant if α = 0. This is the mathematics behind damped and undamped vibrations.

The Three Damping Cases (Second-Order) Given ay″ + by′ + cy = 0 with discriminant Δ = b² − 4ac:

Δ > 0 (overdamped) — two distinct real roots, solution decays without oscillating.
Δ = 0 (critically damped) — one repeated root, solution decays as (C₁ + C₂x)e^rx.
Δ < 0 (underdamped) — complex roots, solution oscillates with exponential envelope.

Laplace Transforms: An Algebraic Bypass

The Laplace transform converts a function of time f(t) into a function of a complex variable s. The crucial fact is that the transform of a derivative is an algebraic expression:

Transform of derivatives

ℒ{y′} = sY(s) − y(0)
ℒ{y″} = s²Y(s) − sy(0) − y′(0)

This converts a differential equation into an algebraic equation in Y(s). Solve for Y(s), then invert back to the time domain using a table of transforms and partial fractions. The Laplace method is especially powerful for initial value problems and for equations with discontinuous or impulsive forcing functions.

Real-World Modelling

Before applying any technique, building a good model matters as much as the algebra. Three classic first-order models appear constantly:

Exponential growth / decay

dN/dt = kN

Population growth (k>0), radioactive decay (k<0), compound interest

Newton's Law of Cooling

dT/dt = k(T − T_env)

Temperature of an object approaching ambient. Separable; solution T(t) = T_env + Ce^kt

RL Circuit

L · dI/dt + RI = V(t)

Current in a series inductor-resistor circuit. First-order linear ODE.

Logistic Growth

dP/dt = rP(1 − P/K)

Population with carrying capacity K. Separable; solution via partial fractions.

§ 06Are You Ready? Prerequisites

Differential equations draw on several earlier topics. Check that you're confident with each before starting.

Must-have Basic Integration Standard integrals, constants of integration, definite vs indefinite. Used in every method. Must-have Integration by Parts Needed for solving certain first-order linear ODEs and for deriving Laplace transforms. Must-have Partial Fractions Essential for separable equations and for inverting Laplace transforms. Helpful Implicit Differentiation Underpins the logic of differentiating equations where y isn't isolated. Helpful Higher-Order Derivatives Second and higher derivatives — central to all second-order ODE work. Background Continuity Existence and uniqueness theorems require understanding of continuity on intervals.

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