Differential Equations and Linear Algebra

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Overview of Differential Equations
Episode overview

Linear equations include dy/dt = y, dy/dt = –y, dy/dt = 2ty. The equation dy/dt = y*y is nonlinear.

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The Calculus You Need
Episode overview

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

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1.4b: Response to Exponential Input, exp(s*t)
Episode overview

With exponential input, e^st, from outside and exponential growth, e^at, from inside, the solution, y(t), is a combination of two exponentials.

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1.4c: Response to Oscillating Input, cos(w*t)
Episode overview

An oscillating input cos(ωt) produces an oscillating output with the same frequency ω (and a phase shift).

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1.4d: Solution for Any Input, q(t)
Episode overview

To solve a linear first order equation, multiply each input q(s) by its growth factor and integrate those outputs.

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1.4e: Step Function and Delta Function
Episode overview

A unit step function jumps from 0 to 1. Its slope is a delta function: zero everywhere except infinite at the jump.

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1.5: Response to Complex Exponential, exp(i*w*t) = cos(w*t)+i*sin(w*t)
Episode overview

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).

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1.6: Integrating Factor for a Constant Rate, a
Episode overview

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

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1.6b: Integrating Factor for a Varying Rate, a(t)
Episode overview

The integral of a varying interest rate provides the exponent in the growing solution (the bank balance).

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1.7: The Logistic Equation
Episode overview

When –by^2 slows down growth and makes the equation nonlinear, the solution approaches a steady state y(∞) = a/b.

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1.7c: The Stability and Instability of Steady States
Episode overview

Steady state solutions can be stable or unstable – a simple test decides.

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1.8: Separable Equations
Episode overview

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.

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2.1: Second Order Equations
Episode overview

For the oscillation equation with no damping and no forcing, all solutions share the same natural frequency

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2.1b: Forced Harmonic Motion
Episode overview

With forcing f = cos(ωt), the particular solution is Y*cos(ωt). But if the forcing frequency equals the natural frequency there is resonance

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2.3: Unforced Damped Motion
Episode overview

With constant coefficients in a differential equation, the basic solutions are exponentials e^st. The exponent s solves a simple equation such as As^2 + Bs + C = 0.

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2.3c: Impulse Response and Step Response
Episode overview

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.

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2.4: Exponential Response - Possible Resonance
Episode overview

Resonance occurs when the natural frequency matches the forcing frequency — equal exponents from inside and outside.

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2.4b: Second Order Equations With Damping
Episode overview

A damped forced equation has a particular solution y = G cos(ωt – α). The damping ratio provides insight into the null solutions.

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2.5: Electrical Networks: Voltages and Currents
Episode overview

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

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2.6: Methods of Undetermined Coefficients
Episode overview

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

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2.6b: An Example of Method of Undetermined Coefficients
Episode overview

This method is also successful for forces and solutions such as (at^2 + bt +c) e^st: substitute into the equation to find a, b, c.

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2.6c: Variations of Parameters
Episode overview

Combine null solutions y1 and y2 with coefficients c1(t) and c2(t) to find a particular solution for any f(t).

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2.7: Laplace Transform: First Order Equation
Episode overview

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).

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2.7b: Laplace Transform: Second Order Equation
Episode overview

The second derivative transforms to s^2Y and the algebra problem involves the transfer function 1/ (As^2 + Bs +C).

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2.7c: Laplace Transforms and Convolution
Episode overview

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.

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3.1: Pictures of the Solutions
Episode overview

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.

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3.2: Phase Plane Pictures: Source, Sink Saddle
Episode overview

Solutions to second order equations can approach infinity or zero. Saddle points contain a positive and also a negative exponent or eigenvalue.

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3.2b: Phase Plane Pictures: Spirals and Centers
Episode overview

Imaginary exponents with pure oscillation provide a “center” in the phase plane. The point (y, dy/dt) travels forever around an ellipse.

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3.2c: Two First Order Equations: Stability
Episode overview

A second order equation gives two first order equations for y and dy/dt. The matrix becomes a companion matrix.

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3.3: Linearization at Critical Points
Episode overview

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.

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3.3b: Linearization of y'=f(y,z) and z'=g(y,z)
Episode overview

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.

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3.3c: Eigenvalues and Stability: 2 by 2 Matrix, A
Episode overview

Two equations y’ = Ay are stable (solutions approach zero) when the trace of A is negative and the determinant is positive.

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3.3d: The Tumbling Box in 3-D
Episode overview

A box in the air can rotate around its shortest and longest axes. Around the middle axis it tumbles wildly.

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5.1: The Column Space of a Matrix, A
Episode overview

An m by n matrix A has n columns each in Rm. Capturing all combinations Av of these columns gives the column space – a subspace of Rm.

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5.4: Independence, Basis, and Dimension
Episode overview

Vectors v1 to vd 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.

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5.5: The Big Picture of Linear Algebra
Episode overview

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.

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5.6: Graphs
Episode overview

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.

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5.6b: Incidence Matrices of Graphs
Episode overview

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.

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6.1: Eigenvalues and Eigenvectors
Episode overview

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

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6.2: Diagonalizing a Matrix
Episode overview

A matrix can be diagonalized if it has n independent eigenvectors. The diagonal matrix Λis the eigenvalue matrix.

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6.2b: Powers, A^n, and Markov Matrices
Episode overview

Diagonalizing A = VΛV–1 also diagonalizes An = VΛnV–1.

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6.3: Solving Linear Systems
Episode overview

dy/dt = Ay contains solutions y = eλtx where λ and x are an eigenvalue / eigenvector pair for A.

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6.4: The Matrix Exponential, exp(A*t)
Episode overview

The shortest form of the solution uses the matrix exponential y = eAt y(0). The matrix eAt has eigenvalues eλt and the eigenvectors of A.

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6.4b: Similar Matrices, A and B=M^(-1)*A*M
Episode overview

A and B are “similar” if B = M-1AM for some matrix M. B then has the same eigenvalues as A.

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6.5: Symmetric Matrices, Real Eigenvalues, Orthogonal Eigenvectors
Episode overview

Symmetric matrices have n perpendicular eigenvectors and n real eigenvalues.

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6.5b: Second Order Systems, y''+Sy=0
Episode overview

An oscillation equation d2y/dt2 + Sy = 0 has 2n solutions (sines and cosines). Solutions use the eigenvectors of S.

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7.2: Positive Definite Matrices, S=A'*A
Episode overview

A positive definite matrix S has positive eigenvalues, positive pivots, positive determinants, and positive energy vTSv for every vector v. S = ATA is always positive definite if A has independent columns.

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7.2b: Singular Value Decomposition, SVD
Episode overview

The SVD factors each matrix A into an orthogonal matrix U times a diagonal matrix Σ (the singular value) times another orthogonal matrix VT: rotation times stretch times rotation.

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7.3: Boundary Conditions Replace Initial Conditions
Episode overview

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).

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7.4: Laplace Equation
Episode overview

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

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8.1: Fourier Series
Episode overview

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

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8.1b: Examples of Fourier Series
Episode overview

Even functions use only cosines (F(–x) = F(x)) and odd functions use only sines. The coefficients an and bn come from integrals of F(x) cos(nx) and F(x) sin(nx).

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8.1c: Fourier Series Solution of Laplace's Equation
Episode overview

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.

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8.3: Heat Equation
Episode overview

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. The heat equation ∂u/∂t = ∂2u/∂x2 starts .. show full overview

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8.4: Wave Equation
Episode overview

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).