Ace Ordinary Differential Equations in 17 Hours

Explore the definitions and terminology of differential equations, distinguishing ordinary differential equations from partial ones, and classify by type, order, and linearity with clear examples.

Verify a function is a solution to a differential equation by differentiating, substituting back, and checking equality, illustrated with two examples: a first-order linear and a linear second-order example.

Solve two separable first-order ODEs, dy/dx = f(x) and dy/dx = g(x) h(y), by separating variables, integrating, and applying initial or boundary conditions for the particular solution.

Explore initial value problems for first and higher order ODEs and how initial conditions determine solutions. Solve a nonlinear first-order ODE by separable variables, yielding y = tan(4x - 3π/4).

Model I covers first-order differential equations with dy/dt proportional to y, solved by separation of variables to give y(t)=y0 e^{alpha t}, exhibiting exponential growth or decay based on alpha.

Explore real life applications of model one by examining population dynamics and radioactive decay, and analyze three numerical examples: bacterial growth, radioactive decay, and carbon dating.

Solve model two using separable variables, obtaining y(t) = n − (n − y0) e^(−kt) for the upper bound and y(t) = n + (y0 − n) e^(−kt) for the lower bound.

Apply Newton's law of cooling to model temperature change with ambient temperature, and the cake example shows solving the separable ode yields a temperature that approaches ambient, roughly 30 minutes.

Explore the logistic equation as a nonlinear first order ODE, derive its separable solution, and note its approach to the upper bound with an inflection point at y = n/2.

Examine model three logistic equation applications with a 1000 upper bound, including disease and rumor spread, and population dynamics, plus a numerical six-day infection example.

Explore five population models based on first-order ODEs, including exponential and logistic growth, and map birth and death dynamics to upper bounds and extinction or steady-state scenarios.

Learn a method to solve first order linear ODEs using integrating factor mu(t) = e^{integral of p(t) dt}, turning the left side into the derivative of mu y.

Apply the integrating factor to solve a first-order linear ODE; derive the homogeneous and particular solutions in Model IV, and show three behaviors: exponential, linear, or decaying-plus-linear based on y0.

This mixture problem models a tank with inflow and outflow, forming dy/dt via concentrations and solving by integrating factor to find the maximum at t=60 and empty at t=200.

Derive the first and second order odes for series LCR circuits using Kirchhoff's law, connecting the inductor, resistor, capacitor, and battery, and apply integrating factors for RC and LR reductions.

apply newtonian gravity to model motion: derive v(x) from a separable differential equation, compute peak altitude and escape velocity sqrt(2 g r) under no air resistance.

Apply newton's second law to a growing hailstone with radius proportional to time, derive a dv/dt + (3/t) v = g, and find constant acceleration a = g/4.

Model a pilot's motion under a constant wind with a homogeneous first-order ode; solve by y/x substitution, derive y(x) in terms of k=w/v0, and discuss k<1, k=1, k>1.

Derive Torricelli's law for a draining fluid, showing v^2/2 = g y and v = sqrt(2 g y) via separable equations, with contraction coefficient c for flow through a hole.

Derive the water-surface area for a hemispherical bowl as A(y)=pi(8y - y^2) and form a separable ODE for y(t); compute empty time t≈3154 s and hole radius r≈0.00444 ft (1.3 cm).

Review the methods I to IV for solving first-order ODEs—direct integration, variable separable, linear with integrating factor, and homogeneous substitution—and note the transition to exact ODE.

Check exactness for m(x,y) dx + n(x,y) dy = 0 via ∂m/∂y = ∂n/∂x, then integrate to build f(x,y) with ∂f/∂x = m and ∂f/∂y = n, so f(x,y)=C.

examine method five for exact differential equations through two examples, verify exactness by comparing partial derivatives, construct the potential function f, and show the solution as f(x,y)=constant.

Apply method six to convert non-exact differential equations into exact ones by using an integrating factor mu. Investigate cases where mu depends on x or y, leading to separable solutions.

Explore method six for turning non-exact first-order odes into exact ones using integrating factors, with two examples and a complete solution via exactness and total derivative.

Analyze Bernoulli's equation and reduce any n to a linear ODE by substituting v = y^{1-n}, yielding dv/dx + (1-n)p(x) v = (1-n) f(x), then recover y = v^{1/(1-n)}.

Solve a Bernoulli equation with a y^2 term by rewriting it to the Bernoulli form (n=2) and substituting v = 1/y. Solve the resulting first-order linear ODE using an integrating factor to obtain y = 1/(c x − x^2) with x ≠ 0.

Master the substitution method for first-order differential equations, including homogeneous and Bernoulli cases, by using a substitution like u = 3x+2y to obtain a separable relation between x and y.

Rewrite the missing-variable equation by setting y' = v(y) and y'' = v dv/dy, reduce to a first-order equation, solve for v, then integrate to y(x) for c>0, c=0, c<0.

Explore initial value problems for nth order linear odes, showing how given y and derivatives at x0 determine constants of integration, with existence and uniqueness guaranteed when coefficients are continuous.

Explore boundary value problems for linear second-order ODEs with constraints at different points. See how boundary conditions form a system that can have no, a unique, or infinitely many solutions.

Verify the general solution of the second-order ode x'' + 16x = 0, then classify boundary value problems under given conditions, revealing infinite, unique, or no solution cases.

Explore homogeneous and non-homogeneous linear nth order odes, define the differential and polynomial operators, and learn how the associated homogeneous equation relates to the non-homogeneous form.

The superposition principle states that linear combinations of two known solutions to a homogeneous linear differential equation, with coefficients c1 and c2, are also solutions on the interval.

Examine linear dependence and independence of functions via linear combinations; analyze a cos^2, sin^2, sec^2, tan^2 example on the interval [-pi/2, pi/2] showing dependence and introduce a testing method.

Use the Wronskian to test linear independence of a fundamental solution set for a homogeneous n-th order ode; a nonzero Wronskian confirms independence and defines the general solution.

Use the Wronskian to test linear independence of homogeneous ode solutions, with two functions e^{3x}, e^{-3x}, and three functions e^{x}, e^{2x}, e^{3x}, to obtain their general solutions.

Form the general solution of a non-homogeneous linear equation by combining the complementary function of the associated homogeneous equation with a particular solution, as shown by yp=3 in y''+9y=27.

Verify given function as a particular solution of a third-order nonhomogeneous equation, and express general solution as complementary function c1 e^x + c2 e^{2x} + c3 e^{3x} plus -11/12 - (1/2)x.

Apply the non-homogeneous superposition principle by summing yp1, yp2, ..., ypk for g1, g2, ..., gk, then form the general solution with the complementary function.

Use reduction of order on second order homogeneous ode with y1 to obtain y2 = y1 ∫ e^{-∫ p(x) dx} / y1^2 and y = k1 y1 + k2 y2.

Apply reduction of order to y1 = e^x, obtaining y2 = e^{-x}; then use the standard formula for y1 = x^2 cos x to get y2 = x^2 sin x.

Explore solving homogeneous second-order linear odes with constant coefficients via the auxiliary equation, covering distinct real, repeated, and complex roots; learn reduction of order and Euler's formula.

Solve homogeneous second-order linear ODEs with constant coefficients using the auxiliary equation, including distinct, repeated, and complex roots; apply to an initial-value problem to determine constants.

Explore two fundamental second-order linear differential equations with constant coefficients: y''+k^2 y=0 and y''-k^2 y=0, solving with cosine, sine, exponential, and hyperbolic functions.

Solve the homogeneous second-order Cauchy-Euler equation a x^2 y'' + b x y' + c y = 0 using the auxiliary equation, yielding real distinct, real repeated, or complex-root solutions.

Solve homogeneous second-order Cauchy-Euler equations by substituting y = x^n, derive the auxiliary equation, and illustrate three examples with distinct real, repeated real, and complex roots, including an initial-value problem.

Solve homogeneous higher-order linear odes with constant coefficients using an exponential trial, derive the auxiliary equation, and form the general solution from real, repeated, and complex roots.

Solve higher order homogeneous linear equations with constant coefficients using extraction of roots and De Moivre's theorem to rewrite complex roots as alpha plus or minus beta i.

Solve higher-order Cauchy–Euler equations using y = x^n; obtain repeated and complex roots, including x and x^2 multipliers, and prepare for non-homogeneous cases.

Use the undetermined coefficients method to find particular solutions of non-homogeneous linear odes with constant coefficients, selecting forms from polynomial, exponential, sine, or cosine and their products.

Explore three examples of the method of undetermined coefficients for non-homogeneous second-order equations with constant coefficients, deriving complementary and particular solutions and combining them.

Explore how to adjust particular solutions using the method of undetermined coefficients when duplication with the complementary solution occurs, by multiplying by x and handling repeated roots.

Use variation of parameters to find a non-homogeneous second-order differential equation's particular solution with variable coefficients, then derive u1 and u2 via the wronskian and Cramer's rule.

Apply variation of parameters to solve two non-homogeneous second-order equations (two examples), derive complementary solutions, compute particular solutions, and obtain the general solution using the wronskian.

Learn to solve second-order non-homogeneous equations using Green's function, constructing the particular solution from y1, y2, and the wronskian for IVPs.

Explore two examples of Green's function for second-order equations, deriving y1 and y2 from the homogeneous equation, forming the Green's function, and obtaining the particular solution via the integral.

Introduce the gamma function, defined by the integral from 0 to infinity of e^{-x} x^{n-1} dx. Derive gamma(n+1)=n gamma(n) via integration by parts, and show that gamma(n+1)=n! for positive integers.

Compute gamma of one half using a substitution and a polar-coordinate evaluation of a double integral. Apply the recursion gamma(n+1)=n gamma(n) to obtain gamma(3/2) and gamma(5/2), with gamma(1/2)=sqrt(pi).

Derive an equation to evaluate integrals of sine and cosine via the gamma function. Convert the double integral to polar coordinates, separate radial and angular parts, and obtain gamma(n) gamma(m)/gamma(n+m).

Apply the gamma function to evaluate cos^n theta sin^m theta integrals, using gamma identities. Work through three examples and use quadrant symmetry to simplify results.

Define the Laplace transform as an integral transform with kernel e^{-st}, derive four transforms (t^n, e^{at}, cosh, cos), and state existence via piecewise continuity and exponential order.

Explore the Laplace transform of derivatives using integration by parts, derive a general pattern for nth derivatives, and apply it to solve initial-value problems with constant-coefficient ODEs.

Explore the first translation theorem for Laplace transforms: translating F(s) by a units corresponds to multiplying f(t) by e^{a t}, with examples using t^3 and e^{-2t} cos 4t to illustrate.

Solve two initial value problems with initial conditions for non-homogeneous second-order ODEs with constant coefficients using the three-step Laplace transform method and the first translation theorem.

Learn the second rule of Laplace transforms: the transform of an integral equals F(s)/s, with g(t) as the integral of f(t) and g'(t)=f(t), plus the inverse form.

Apply transforms of integrals to solve differential equations using Laplace methods, derive inverse Laplace rules for f(s)/s, and illustrate with examples, including a solution y(t)=t e^{-3t} via the translation theorem.

Derive the Laplace transform derivative and its rule for t^n f(t) using Leibniz, then apply to a variable-coefficient differential equation to obtain y(t)=sin t / t.

derive the fourth rule for integrals of transforms and show the Laplace of f(t)/t equals the integral from s to infinity of F(u) du.

Show how to compute Laplace transforms of f(t)/t using a derived rule, with examples sine t over t and (e^{-8t}-e^{-t})/t, linking limits to pi/2.

Explore the second translation theorem for the Laplace transform using the unit step function u(t−a) to shift functions in time. Show that L{f(t−a)u(t−a)} = e^{−as}F(s) and present the inverse form.

Analyze a series rc circuit driven by a step voltage using unit step functions and Laplace transforms to derive i(t) as exponential for 0–a, a–b, and t > b.

Learn the convolution theorem and how Laplace transform converts convolution to product. See the inverse form: inverse Laplace of F(s)G(s) equals f * g under piecewise continuity and exponential order.

Apply convolution theorem to solve differential equations by turning a product in s into convolution of f and g, with f=g=1/(s^2+k^2) using the inverse Laplace to obtain time-domain solution.

Relate Green's function to the convolution theorem to solve a non-homogeneous linear second-order initial-value problem with given initial conditions using Laplace transforms, via convolution.

Explore two Green's function applications for solving nonhomogeneous ODEs via the convolution theorem, including complementary and particular solutions, Laplace transforms, and integral representations.

Apply the convolution theorem to solve Volterra integral equations by taking the Laplace transform, isolating y in the s-domain, and returning to the t-domain to obtain y(t).

Use the convolution theorem to evaluate integrals by convolving f and g with f(t)=t^{n-1} and g(t)=t^{m-1}, derive the beta function via gamma functions, and illustrate with two examples.

Derives the Laplace transform for periodic functions with period T, proving L{f}(s)=1/(1−e^{−sT}) ∫_0^T e^{−st} f(t) dt, with a two-second pulse example.

Explain how the Laplace transform handles the Dirac delta as a limit of the unit impulse, with area one and e^{-s t0}, and apply to an impulse-based initial value problem.

Represent a first-order linear system with x' = Ax + f(t) in matrix form, compare homogeneous and non-homogeneous cases, and apply initial conditions for a unique solution.

Explore theorems for homogeneous and non-homogeneous systems, including the superposition principle and linear independence. Learn to form the general solution from a fundamental set of solutions and a particular solution.

Verify solutions of a linear system and derive the general solution of a non-homogeneous system using a fundamental set and a particular solution, and verify homogeneous part via the wronskian.

Derive solutions for homogeneous linear systems with constant coefficients using x' = A x, and identify eigenvalues and eigenvectors from the characteristic equation det(A - lambda I) = 0.

Use distinct real eigenvalues and eigenvectors to solve a homogeneous linear system; express x(t) as a sum of eigenvector terms e^{lambda t}, as shown with lambda -1 and 4.

Explore repeated eigenvalues in linear systems by constructing generalized eigenvectors and solutions with e^(lambda t) and t-powers, using p and q vectors from examples.

Explore how complex eigenvalues appear in conjugate pairs and, via Euler's formula, yield real solutions using b1 and b2 as the eigenvector parts, with e^{alpha t} and trig terms.

Apply the undetermined coefficients method to nonhomogeneous linear systems with constant coefficients, find the complementary function via eigenvalues and eigenvectors, determine a particular solution, and obtain the general solution.

Explore variation of parameters for non-homogeneous linear systems, using the fundamental matrix Phi(t) and its inverse to form a particular solution Phi(t) u(t), with u'(t) = Phi^{-1}(t) f(t).

Solve a non-homogeneous linear system using variation of parameters, finding the complementary function and a particular solution, then combine them for the general solution.

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