🌀Differential Equations

What it means for a function to be the unknown, and how a direction field shows every solution before you solve anything.

1.1 What a Differential Equation Actually Is · 1.1 What It Means to Solve a Differential Equation · 1.1 General Solutions Versus Particular Solutions · 1.2 Verifying That a Function Solves a Differential Equation · +4 more

The toolbox for a first-order equation: separating variables, integrating factors, exactness, and the substitutions that turn a hard equation into an easy one.

2.1 Separable Equations · 2.2 Linear First-Order Equations and the Integrating Factor · 2.3 Exact Equations · 2.4 Turning a Non-Exact Equation into an Exact One · +3 more

When a solution is guaranteed to exist and be the only one, and how a phase line predicts a solution's fate without ever solving the equation.

2.8 Existence and Uniqueness for First-Order Equations · 2.8 Why Solution Curves of the Same Equation Never Cross · 2.8 When the Uniqueness Theorem's Conditions Fail · 2.5 Autonomous Equations and Equilibrium Solutions · +4 more

Growth, cooling, mixing, and falling with drag, all written down as first-order equations and solved with real numbers.

2.3 Exponential Growth and Decay Models · 2.3 Half-Life and Doubling Time · 2.3 The Logistic Model of Population Growth · 2.3 Setting Up a Mixing Problem · +4 more

What to do when no formula exists: stepping a solution forward numerically, and watching the error shrink as the steps get smaller.

8.1 Why Numerical Methods Matter · 8.1 Euler's Method · 8.1 Stepping Through Euler's Method by Hand · 8.1 Local and Global Error in Euler's Method · +2 more

Guessing y = e^(rt) turns calculus into algebra: the characteristic equation, its three root cases, and how the Wronskian proves two solutions are enough.

3.1 Second-Order Linear Equations and Homogeneity · 3.2 The Principle of Superposition · 3.1 The Characteristic Equation · 3.1 Real and Distinct Roots · +6 more

Forcing terms need a particular solution on top of the homogeneous one: guessing it with undetermined coefficients, or building it with variation of parameters.

3.5 The General Solution of a Nonhomogeneous Equation · 3.5 Undetermined Coefficients for Polynomial and Exponential Forcing · 3.5 Undetermined Coefficients for Sine and Cosine Forcing · 3.5 The Resonant Case in Undetermined Coefficients · +4 more

A mass on a spring, damped, forced, and pushed to resonance — the three root cases from Unit 6 made physical, plus the RLC circuit that obeys the same equation.

3.7 Setting Up the Spring-Mass Equation · 3.7 Free Undamped Vibrations · 3.7 Solving a Free Undamped Vibration Problem · 3.8 The Three Cases of Free Damped Vibration · +6 more

Turning a differential equation into algebra: transforms, inverses, initial value problems in one pass, step and impulse forcing, and convolution.

6.1 What the Laplace Transform Does · 6.1 The Definition of the Laplace Transform · 6.1 Transforms of Common Functions · 6.1 Linearity of the Laplace Transform · +10 more

What to do when an equation like Airy's has no elementary solution: building one term by term as a power series instead.

5.1 Why Some Equations Need a Series Solution · 5.2 Series Solutions Near an Ordinary Point · 5.3 Ordinary Points and Singular Points · 5.4 The Euler Equidimensional Equation · +4 more

Where differential equations and linear algebra become one subject: eigenvalues and eigenvectors as the straight-line solutions of a whole system.

7.1 Converting a Higher-Order Equation into a System · 7.1 Writing a System in Matrix Form · 7.3 Eigenvalues, Eigenvectors, and Straight-Line Solutions · 7.5 Solving a System with Real, Distinct Eigenvalues · +6 more

Trajectories flowing in the phase plane, equilibria read straight off eigenvalues, and what linearization can and can't tell you about a nonlinear system.

9.1 The Phase Plane · 9.1 Classifying Equilibria from Eigenvalues · 9.1 Nodes · 9.1 Saddle Points · +5 more

One equation splits into two ordinary ones, a square wave gets built from harmonics, and heat diffuses flat while a wave keeps traveling.

10.1 What Makes an Equation Partial Instead of Ordinary · 10.5 Separation of Variables · 10.5 Why Separation of Variables Doesn't Always Work · 10.2 Building a Periodic Function from Sines and Cosines · +8 more