Differential Equations

Units

1. 1

   First-order equations

   8 lessons

   1. Modelling situations with differential equations and verifying solutionsTranslate a rate description into a differential equation, verify a function, and use a condition to choose a solution.
   2. Sketching and reasoning with slope fieldsSketch slope segments, trace a solution through a given point, and recognize patterns and equilibria.
   3. Approximating solutions using Euler's methodEstimate a solution with Euler’s method and justify whether the estimate is above or below the actual value.
   4. General and particular solutions by separation of variablesFind a general solution, use an initial condition to select a particular solution, and state its largest domain.
   5. Exponential models with differential equationsSolve proportional-rate equations, use half-life or doubling time, and interpret a nonzero baseline.
   6. First-order linear equations and integrating factorsSolve y' + P(x) y = Q(x) by the integrating factor, with the constant in the right place, and fit an initial condition.
   7. Modelling with first-order equations: cooling, mixing and orthogonal trajectoriesTurn a cooling story, a mixing tank (fixed or changing volume) and a family of curves into first-order equations, solve them and check the results.
   8. Logistic models with differential equationsFind a logistic model’s limit, fastest-growth population, and solution from an initial value.
2. 2

   Exact equations and qualitative theory

   3 lessons

   1. Exact equationsRecognize an exact equation M dx + N dy = 0 by the test ∂M/∂y = ∂N/∂x, find its potential ψ, and use an integrating factor when the test fails.
   2. Existence and uniquenessState the existence and uniqueness theorem for y' = f(t, y), check its conditions, and find the interval on which a solution of a linear equation is guaranteed.
   3. Autonomous equations and phase linesFor y′ = f(y), find the equilibria as zeros of f, classify each as stable, unstable or semistable from the signs of f or the slope f′, draw the phase line, and follow how a harvesting term moves and merges the equilibria.
3. 3

   Hyperbolic functions and complex exponentials

   3 lessons

   1. Hyperbolic functionsDefine cosh x and sinh x from the exponentials, graph them, show cosh²x − sinh²x = 1 as the hyperbola they trace, and use cosh to describe a hanging cable.
   2. Calculus of hyperbolic functionsDifferentiate and integrate sinh, cosh and tanh, write the inverse hyperbolic functions as logarithms, and evaluate ∫ dx/√(x² + 1) = sinh⁻¹ x + C.
   3. Complex numbers and Euler's formulaWrite a complex number as a point and as r e^{iθ}, prove e^{iθ} = cos θ + i sin θ from the Taylor series, and use it to multiply by rotating and scaling and to raise to powers.
4. 4

   Second-order linear equations

   4 lessons

   1. Linear second-order equations and the WronskianShow that combinations of solutions of y'' + p y' + q y = 0 are solutions, that two solutions form a fundamental set when their Wronskian is nonzero, and that c₁y₁ + c₂y₂ then fits any initial position and velocity.
   2. Constant coefficients: distinct real rootsSolve ay'' + by' + cy = 0 by trying y = e^{rt}, find the general solution when the characteristic equation has two distinct real roots, and fit it to y(0) and y'(0).
   3. Complex and repeated rootsSolve ay'' + by' + cy = 0 when the characteristic equation has complex roots λ ± μi or one repeated root, and compare the three cases on one set of axes.
   4. Mechanical vibrationsModel a mass on a spring with damping by my'' + γy' + ky = 0, find the natural frequency, amplitude and phase of undamped motion, and classify damped motion as underdamped, critically damped or overdamped.
5. 5

   Forced equations

   3 lessons

   1. Undetermined coefficientsSolve ay'' + by' + cy = g(t) as y = y_c + y_p, guess the form of a particular solution y_p for polynomial, exponential, sine and cosine forcing, and multiply by t when the guess already solves the homogeneous equation.
   2. Variation of parametersFind a particular solution y_p = u₁y₁ + u₂y₂ of y'' + p y' + q y = g(t) for any continuous g from the Wronskian formulas, and use it where undetermined coefficients fails.
   3. Forced oscillations and resonanceSolve y'' + γy' + ky = F₀ cos ωt for a spring driven at rate ω, explain beats when ω is near the natural rate, unbounded growth at resonance without damping, and read the steady-state amplitude curve against ω with damping.
6. 6

   Linear systems and phase portraits

   5 lessons

   1. Systems of equations in matrix formWrite two coupled equations, such as two tanks of brine or a second-order equation, as x' = Ax, read a solution as a curve in the phase plane over the direction field, and check that x = e^{λt}v solves it when Av = λv.
   2. Real eigenvalues: nodes and saddlesSolve x' = Ax when A has two distinct real eigenvalues, and read the phase portrait (saddle or node) from their signs.
   3. Complex eigenvalues: spirals and centresSolve x' = Ax when A has eigenvalues λ ± μi, write the real solutions e^{λt} times rotations, and read the phase portrait as a spiral sink, a spiral source or a centre, with the direction of turning from one arrow of the field.
   4. Repeated eigenvaluesSolve x' = Ax when A has one repeated eigenvalue and only one eigenvector, build the second solution te^{λt}v + e^{λt}w from a generalized eigenvector w, and sort every 2 × 2 case on the trace-determinant plane.
   5. Nonlinear systems and linearizationFind the equilibria of a nonlinear system x' = P(x, y), y' = Q(x, y), compute the Jacobian at each, read the local portrait (saddle, centre or spiral) from its eigenvalues, and stitch the local pictures into the whole phase portrait.
7. 7

   Series solutions

   2 lessons

   1. Power series solutions near an ordinary pointSolve a linear equation y'' + p(x)y' + q(x)y = 0 with polynomial coefficients by writing y = Σ aₙxⁿ, shifting indices to get a recurrence for aₙ, and reading off the solution with its two free constants.
   2. Euler equations and the Frobenius methodSolve an Euler equation x²y'' + αxy' + βy = 0 with y = xʳ, and at a regular singular point write the solution as xʳ Σ aₙxⁿ, with the indicial equation choosing r.
8. 8

   Laplace transforms

   4 lessons

   1. The Laplace transformCompute L{f}(s) = ∫₀^∞ e^{−st} f(t) dt as an improper integral for 1, e^{at}, sin bt and t, use linearity, and derive the derivative rule L{y'} = sY − y(0).
   2. Solving initial value problemsTransform a linear constant-coefficient initial value problem into an algebraic equation for Y(s), split Y(s) by partial fractions and invert each piece with the table.
   3. Step functions and discontinuous forcingWrite a switched-on or pulsed forcing with unit steps u_c(t), transform it with L{u_c} = e^{−cs}/s and the shift rule L{u_c(t)f(t − c)} = e^{−cs}F(s), and solve a spring driven by a pulse.
   4. Impulses and convolutionModel a hammer blow with the Dirac delta, use L{δ(t − c)} = e^{−cs} to solve a spring struck once, and write the response to any forcing as the convolution of the forcing with the impulse response.