1.1: Overview of Differential Equations Linear equations include dy/dt = y, dy/dt = –y, dy/dt = 2ty. The equation dy/dt = y*y is nonlinear.

1.2: The Calculus You Need The sum rule, product rule, and chain rule produce new derivatives from the derivatives of xⁿ, sin(x) and e^x. The Fundamental Theorem of Calculus says that the integral inverts the derivative.

1.4b: Response to Exponential Input, exp(s*t) With exponential input, e^st, from outside and exponential growth, e^at, from inside, the solution, y(t), is a combination of two exponentials.

1.4c: Response to Oscillating Input, cos(w*t) An oscillating input cos(ωt) produces an oscillating output with the same frequency ω (and a phase shift).

1.4d: Solution for Any Input, q(t) To solve a linear first order equation, multiply each input q(s) by its growth factor and integrate those outputs.

1.4e: Step Function and Delta Function A unit step function jumps from 0 to 1. Its slope is a delta function: zero everywhere except infinite at the jump.

1.5: Response to Complex Exponential, exp(i*w*t) = cos(w*t)+i*sin(w*t) For linear equations, the solution for f = cos(ωt) is the real part of the solution for f = e^iωt. That complex solution has magnitude G (the gain).

1.6: Integrating Factor for a Constant Rate, a The integrating factor e^-at multiplies the differential equation, y’=ay+q, to give the derivative of e^-aty: ready for integration.

1.6b: Integrating Factor for a Varying Rate, a(t) The integral of a varying interest rate provides the exponent in the growing solution (the bank balance).

1.7: The Logistic Equation When –by² slows down growth and makes the equation nonlinear, the solution approaches a steady state y(∞) = a/b.

1.7c: The Stability and Instability of Steady States Steady state solutions can be stable or unstable – a simple test decides.

1.8: Separable Equations Separable equations can be solved by two separate integrations, one in t and the other in y. The simplest is dy/dt = y, when dy/y equals dt. Then ln(y) = t + C.

2.1: Second Order Equations For the oscillation equation with no damping and no forcing, all solutions share the same natural frequency.

2.1b: Forced Harmonic Motion With forcing f = cos(ωt), the particular solution is Y*cos(ωt). But if the forcing frequency equals the natural frequency there is resonance.

2.3: Unforced Damped Motion With constant coefficients in a differential equation, the basic solutions are exponentials e^st. The exponent s solves a simple equation such as As² + Bs + C = 0.

2.3c: Impulse Response and Step Response The impulse response g is the solution when the force is an impulse (a delta function). This also solves a null equation (no force) with a nonzero initial condition.

2.4: Exponential Response - Possible Resonance Resonance occurs when the natural frequency matches the forcing frequency — equal exponents from inside and outside.

2.4b: Second Order Equations With Damping A damped forced equation has a particular solution y = G cos(ωt – α). The damping ratio provides insight into the null solutions.

2.5: Electrical Networks: Voltages and Currents Current flowing around an RLC loop solves a linear equation with coefficients L (inductance), R (resistance), and 1/C (C = capacitance).

2.6: Methods of Undetermined Coefficients With constant coefficients and special forcing terms (powers of t, cosines/sines, exponentials), a particular solution has this same form.

2.6b: An Example of Method of Undetermined Coefficients This method is also successful for forces and solutions such as (at² + bt +c) e^st: substitute into the equation to find a, b, c.

2.6c: Variations of Parameters Combine null solutions y₁ and y₂ with coefficients c₁(t) and c₂(t) to find a particular solution for any f(t).

2.7: Laplace Transform: First Order Equation Transform each term in the linear differential equation to create an algebra problem. You can then transform the algebra solution back to the ODE solution, y(t).

2.7b: Laplace Transform: Second Order Equation The second derivative transforms to s²Y and the algebra problem involves the transfer function 1/ (As² + Bs +C).

2.7c: Laplace Transforms and Convolution When the force is an impulse δ (t), the impulse response is g(t). When the force is f(t), the response is the “convolution” of f and g.

3.1: Pictures of the Solutions The direction field for dy/dt = f(t,y) has an arrow with slope f at each point t, y. Arrows with the same slope lie along an isocline.

3.2: Phase Plane Pictures: Source, Sink Saddle Solutions to second order equations can approach infinity or zero. Saddle points contain a positive and also a negative exponent or eigenvalue.

3.2b: Phase Plane Pictures: Spirals and Centers Imaginary exponents with pure oscillation provide a “center” in the phase plane. The point (y, dy/dt) travels forever around an ellipse.

3.2c: Two First Order Equations: Stability A second order equation gives two first order equations for y and dy/dt. The matrix becomes a companion matrix.

3.3: Linearization at Critical Points A critical point is a constant solution Y to the differential equation y’ = f(y). Near that Y, the sign of df/dy decides stability or instability.

3.3b: Linearization of y'=f(y,z) and z'=g(y,z) With two equations, a critical point has f(Y,Z) = 0 and g(Y,Z) = 0. Near those constant solutions, the two linearized equations use the 2 by 2 matrix of partial derivatives of f and g.

3.3c: Eigenvalues and Stability: 2 by 2 Matrix, A Two equations y’ = Ay are stable (solutions approach zero) when the trace of A is negative and the determinant is positive.

3.3d: The Tumbling Box in 3-D A box in the air can rotate around its shortest and longest axes. Around the middle axis it tumbles wildly.

5.1: The Column Space of a Matrix, A An m by n matrix A has n columns each in R^m. Capturing all combinations Av of these columns gives the column space – a subspace of R^m.

5.4: Independence, Basis, and Dimension Vectors v 1 to v d are a basis for a subspace if their combinations span the whole subspace and are independent: no basis vector is a combination of the others. Dimension d = number of basis vectors.

5.5: The Big Picture of Linear Algebra A matrix produces four subspaces – column space, row space (same dimension), the space of vectors perpendicular to all rows (the nullspace), and the space of vectors perpendicular to all columns.

5.6: Graphs A graph has n nodes connected by m edges (other edges can be missing). This is a useful model for the Internet, the brain, pipeline systems, and much more.

5.6b: Incidence Matrices of Graphs The incidence matrix A has a row for every edge, containing -1 and +1 to show the two nodes (two columns of A) that are connected by that edge.

6.1: Eigenvalues and Eigenvectors The eigenvectors x remain in the same direction when multiplied by the matrix (Ax = λx). An n x n matrix has n eigenvalues.

6.2: Diagonalizing a Matrix A matrix can be diagonalized if it has n independent eigenvectors. The diagonal matrix Λis the eigenvalue matrix.

6.2b: Powers, A^n, and Markov Matrices Diagonalizing A = VΛV^–1 also diagonalizes Aⁿ = VΛⁿV^–1.

6.3: Solving Linear Systems dy/dt = Ay contains solutions y = e^λtx where λ and x are an eigenvalue / eigenvector pair for A.

6.4: The Matrix Exponential, exp(A*t) The shortest form of the solution uses the matrix exponential y = e^At y(0). The matrix e^At has eigenvalues e^λt and the eigenvectors of A.

6.4b: Similar Matrices, A and B=M^(-1)*A*M A and B are “similar” if B = M^-1AM for some matrix M. B then has the same eigenvalues as A.

6.5: Symmetric Matrices, Real Eigenvalues, Orthogonal Eigenvectors Symmetric matrices have n perpendicular eigenvectors and n real eigenvalues.

6.5b: Second Order Systems, y''+Sy=0 An oscillation equation d²y/dt² + Sy = 0 has 2n solutions (sines and cosines). Solutions use the eigenvectors of S.

7.2: Positive Definite Matrices, S=A'*A A positive definite matrix S has positive eigenvalues, positive pivots, positive determinants, and positive energy v^TSv for every vector v. S = A^TA is always positive definite if A has independent columns.

7.2b: Singular Value Decomposition, SVD The SVD factors each matrix A into an orthogonal matrix U times a diagonal matrix Σ (the singular value) times another orthogonal matrix V^T: rotation times stretch times rotation.

7.3: Boundary Conditions Replace Initial Conditions A second order equation can change its initial conditions on y(0) and dy/dt(0) to boundary conditions on y(0) and y(1).

7.4: Laplace Equation The partial differential equation ∂²u/∂x² + ∂²u/∂y² = 0 describes temperature distribution inside a circle or a square or any plane region.

8.1: Fourier Series A Fourier series separates a periodic function F(x) into a combination (infinite) of all basis functions cos(nx) and sin(nx).

8.1b: Examples of Fourier Series Even functions use only cosines (F(–x) = F(x)) and odd functions use only sines. The coefficients a_n and b_n come from integrals of F(x) cos(nx) and F(x) sin(nx).

8.1c: Fourier Series Solution of Laplace's Equation Inside a circle, the solution u(r, θ) combines rⁿ cos(nθ) and rⁿ sin(nθ). The boundary solution combines all entries in a Fourier series to match the boundary conditions.

8.3: Heat Equation The heat equation ∂u/∂t = ∂²u/∂x² starts from a temperature distribution u at t = 0 and follows it for t > 0 as it quickly becomes smooth.

8.4: Wave Equation The wave equation ∂²u/∂t² = ∂²u/∂x² shows how waves move along the x axis, starting from a wave shape u(0) and its velocity ∂u/∂t(0).