Differential Equations on Teddy Rook Bookhttps://teddyrookbook.com/math/differential-equations/Recent content in Differential Equations on Teddy Rook BookTeddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenOrder, Linearity and Solutionshttps://teddyrookbook.com/math/differential-equations/classify-des/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/classify-des/Classify a differential equation and check a proposed solution. Differential Equations · Math LadderInitial Value Problems, Existence and Uniquenesshttps://teddyrookbook.com/math/differential-equations/existence-uniqueness-de/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/existence-uniqueness-de/Know when exactly one solution passes through a point. Differential Equations · Math LadderLinear Equations and Integrating Factorshttps://teddyrookbook.com/math/differential-equations/linear-first-order/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/linear-first-order/Multiply by e^(∫p dt) so the left side becomes a product's derivative. Differential Equations · Math LadderExact Equationshttps://teddyrookbook.com/math/differential-equations/exact-equations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/exact-equations/M dx + N dy = 0 is exact when My = Nx. Differential Equations · Math LadderHomogeneous and Bernoulli Equationshttps://teddyrookbook.com/math/differential-equations/substitutions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/substitutions/Change variables to reach a separable or linear equation. Differential Equations · Math LadderAutonomous Equations and the Phase Linehttps://teddyrookbook.com/math/differential-equations/autonomous-phase-line/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/autonomous-phase-line/Equilibria and their stability come from the sign of f(y). Differential Equations · Math LadderImproved Euler and Runge–Kutta Methodshttps://teddyrookbook.com/math/differential-equations/numerical-methods/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/numerical-methods/Average slopes for much better accuracy. Differential Equations · Math LadderMixing Problemshttps://teddyrookbook.com/math/differential-equations/mixing-problems/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/mixing-problems/Rate of change = rate in − rate out. Differential Equations · Math LadderGrowth, Decay and Newton's Law of Coolinghttps://teddyrookbook.com/math/differential-equations/growth-cooling/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/growth-cooling/Rate proportional to amount, or to the temperature difference. Differential Equations · Math LadderPopulation Models with Harvestinghttps://teddyrookbook.com/math/differential-equations/population-harvesting/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/population-harvesting/Add harvesting to logistic growth and find the new equilibria. Differential Equations · Math LadderFalling Objects with Drag and RC Circuitshttps://teddyrookbook.com/math/differential-equations/motion-circuits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/motion-circuits/Model velocity with air resistance and charging circuits. Differential Equations · Math LadderSecond-Order Linear Equations and Superpositionhttps://teddyrookbook.com/math/differential-equations/second-order-intro/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/second-order-intro/Combinations of solutions of a homogeneous linear equation are solutions. Differential Equations · Math LadderIndependence and the Wronskianhttps://teddyrookbook.com/math/differential-equations/wronskian/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/wronskian/Two solutions form a basis when their Wronskian is not zero. Differential Equations · Math LadderConstant Coefficients: Distinct Real Rootshttps://teddyrookbook.com/math/differential-equations/real-roots/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/real-roots/Try y = e^(rt) and solve the characteristic equation. Differential Equations · Math LadderConstant Coefficients: Complex Rootshttps://teddyrookbook.com/math/differential-equations/complex-roots-de/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/complex-roots-de/e^(αt)(c₁ cos βt + c₂ sin βt) from Euler's formula. Differential Equations · Math LadderConstant Coefficients: Repeated Rootshttps://teddyrookbook.com/math/differential-equations/repeated-roots/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/repeated-roots/A second solution is te^(rt). Differential Equations · Math LadderReduction of Orderhttps://teddyrookbook.com/math/differential-equations/reduction-of-order/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/reduction-of-order/Use one known solution to find another. Differential Equations · Math LadderUndetermined Coefficientshttps://teddyrookbook.com/math/differential-equations/undetermined-coefficients/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/undetermined-coefficients/Guess a particular solution shaped like the forcing term. Differential Equations · Math LadderVariation of Parametershttps://teddyrookbook.com/math/differential-equations/variation-of-parameters/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/variation-of-parameters/A general method for any forcing term. Differential Equations · Math LadderFree Vibrations and Dampinghttps://teddyrookbook.com/math/differential-equations/free-vibrations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/free-vibrations/Underdamped, critically damped and overdamped springs. Differential Equations · Math LadderForced Vibrations and Resonancehttps://teddyrookbook.com/math/differential-equations/forced-resonance/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/forced-resonance/Driving near the natural frequency makes the amplitude grow. Differential Equations · Math LadderRLC Circuitshttps://teddyrookbook.com/math/differential-equations/rlc-circuits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/rlc-circuits/The same equation as a damped spring, in electrical form. Differential Equations · Math LadderHigher-Order Linear Equationshttps://teddyrookbook.com/math/differential-equations/higher-order/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/higher-order/Extend the characteristic-equation method to order n. Differential Equations · Math LadderThe Laplace Transformhttps://teddyrookbook.com/math/differential-equations/laplace-definition/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/laplace-definition/L{f}(s) = ∫₀^∞ e^(−st) f(t) dt. Differential Equations · Math LadderTransforms of Derivatives and Shiftshttps://teddyrookbook.com/math/differential-equations/laplace-properties/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/laplace-properties/L{y′} = sY − y(0), and the shift rules. Differential Equations · Math LadderInverse Transformshttps://teddyrookbook.com/math/differential-equations/inverse-laplace/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/inverse-laplace/Use partial fractions and tables to go back to t. Differential Equations · Math LadderSolving Initial Value Problems with Laplacehttps://teddyrookbook.com/math/differential-equations/laplace-ivps/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/laplace-ivps/Transform, solve for Y, and invert. Differential Equations · Math LadderStep Functions and Piecewise Forcinghttps://teddyrookbook.com/math/differential-equations/step-forcing/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/step-forcing/Use u(t − c) to switch forces on and off. Differential Equations · Math LadderImpulses and the Delta Functionhttps://teddyrookbook.com/math/differential-equations/impulses/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/impulses/Model a sudden kick with δ(t − c). Differential Equations · Math LadderConvolutionhttps://teddyrookbook.com/math/differential-equations/convolution/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/convolution/The transform of a convolution is a product of transforms. Differential Equations · Math LadderSystems in Matrix Formhttps://teddyrookbook.com/math/differential-equations/systems-matrix-form/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/systems-matrix-form/Write x′ = Ax, and rewrite a second-order equation as a system. Differential Equations · Math LadderSolving Systems with Real Eigenvalueshttps://teddyrookbook.com/math/differential-equations/eigen-solutions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/eigen-solutions/x = c₁e^(λ₁t)v₁ + c₂e^(λ₂t)v₂. Differential Equations · Math LadderPhase Portraitshttps://teddyrookbook.com/math/differential-equations/phase-portraits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/phase-portraits/Nodes, saddles, spirals and centers from the eigenvalues. Differential Equations · Math LadderSystems with Complex Eigenvalueshttps://teddyrookbook.com/math/differential-equations/complex-eigen-systems/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/complex-eigen-systems/Real and imaginary parts give spiraling solutions. Differential Equations · Math LadderSystems with Repeated Eigenvalueshttps://teddyrookbook.com/math/differential-equations/repeated-eigen-systems/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/repeated-eigen-systems/Find a generalized eigenvector when one is missing. Differential Equations · Math LadderThe Matrix Exponentialhttps://teddyrookbook.com/math/differential-equations/matrix-exponential/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/matrix-exponential/x(t) = e^(At)x(0). Differential Equations · Math LadderNonhomogeneous Systemshttps://teddyrookbook.com/math/differential-equations/nonhomogeneous-systems/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/nonhomogeneous-systems/Add a particular solution to the homogeneous solution. Differential Equations · Math LadderNonlinear Systems and Linearizationhttps://teddyrookbook.com/math/differential-equations/linearization/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/linearization/Classify equilibria with the Jacobian. Differential Equations · Math LadderPredator–Prey and the Pendulumhttps://teddyrookbook.com/math/differential-equations/predator-prey-pendulum/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/predator-prey-pendulum/Study classic nonlinear models in the phase plane. Differential Equations · Math LadderPower Series Solutions near Ordinary Pointshttps://teddyrookbook.com/math/differential-equations/series-solutions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/series-solutions/Substitute y = Σaₙxⁿ and find a recurrence. Differential Equations · Math LadderCauchy–Euler Equationshttps://teddyrookbook.com/math/differential-equations/cauchy-euler/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/cauchy-euler/Try y = xʳ for x²y″ + axy′ + by = 0. Differential Equations · Math LadderRegular Singular Points and Frobeniushttps://teddyrookbook.com/math/differential-equations/frobenius/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/frobenius/Use y = Σaₙx^(n + r) near a regular singular point. Differential Equations · Math LadderBoundary Value Problems and Eigenfunctionshttps://teddyrookbook.com/math/differential-equations/boundary-value-problems/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/boundary-value-problems/Some values of λ allow nonzero solutions with fixed ends. Differential Equations · Math LadderFourier Serieshttps://teddyrookbook.com/math/differential-equations/fourier-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/fourier-series/Write a periodic function as a sum of sines and cosines. Differential Equations · Math LadderFourier Sine and Cosine Series and Convergencehttps://teddyrookbook.com/math/differential-equations/sine-cosine-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/sine-cosine-series/Extend a function oddly or evenly; the series meets jumps halfway. Differential Equations · Math LadderSeparation of Variableshttps://teddyrookbook.com/math/differential-equations/separation-of-variables/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/separation-of-variables/Split u(x, t) = X(x)T(t) to get ODEs. Differential Equations · Math LadderThe Heat Equationhttps://teddyrookbook.com/math/differential-equations/heat-equation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/heat-equation/Solve heat flow in a rod with Fourier series. Differential Equations · Math LadderThe Wave Equationhttps://teddyrookbook.com/math/differential-equations/wave-equation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/wave-equation/Model a vibrating string. Differential Equations · Math LadderLaplace's Equationhttps://teddyrookbook.com/math/differential-equations/laplace-equation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/differential-equations/laplace-equation/Find steady-state temperatures in a rectangle. Differential Equations · Math Ladder