Become a Differential Equations Master

Explore first order differential equations, including linear, separable, and Bernoulli forms, and map their relation to the highest order derivative and the course roadmap.

Learn to classify differential equations by order, linearity, and homogeneity. Distinguish ordinary differential equations from partial differential equations and identify linear, nonlinear, homogeneous, and nonhomogeneous forms.

Learn to solve first order linear differential equations by matching to the standard form, distinguishing homogeneous and non homogeneous cases, and applying the integrating factor method with a worked example.

Determine the general solution of a first-order linear differential equation with the constant of integration, then use the initial condition to get the specific solution that passes through a point.

Learn how to solve separable differential equations by separating variables, integrating both sides, and consolidating constants to obtain the general solution y = 1/(cos x - C).

Learn to convert non-separable differential equations by substituting u = a x + b y, turning them into linear or separable forms, then solve with integrating factor or separation.

Explore how Bernoulli equations convert to linear form via substitution y^(1-n); solve cases n=0, n=1, and others, using integrating factor to find the general solution y = [3x^2/(1+c x^{3/2})]^{1/3}.

Identify and solve homogeneous differential equations by rewriting in terms of y/x, using v = y/x to obtain a separable equation, then back-substitute to the general solution.

Identify exact differential equations by verifying ∂M/∂y equals ∂N/∂x, then solve by integrating M with respect to x or N with respect to y and finding H(y) or H(x).

Explore linear second-order differential equations, starting with homogeneous equations and reduction of order, then extend to non-homogeneous cases using undetermined coefficients and variation of parameters.

This lecture introduces second order linear homogeneous differential equations with constant coefficients, builds the characteristic equation, classifies root types, and derives the corresponding general solutions.

Apply reduction of order to find the second solution of a second-order linear differential equation from a known solution; set y2 = v y1 and derive y2 = e^{-x/2}.

use undetermined coefficients to solve nonhomogeneous second-order equations by combining the complementary homogeneous solution with a polynomial and exponential particular solution, removing overlaps, and solving for constants.

Apply variation of parameters to solve a second-order nonhomogeneous differential equation by deriving the complementary solution from the homogeneous equation and building a particular solution from u1 and u2.

Demonstrate that a nonzero Wronskian guarantees a fundamental, linearly independent solution set for second-order differential equations, using initial conditions 1001 to build two independent solutions and the general solution form.

Learn variation of parameters for a second-order nonhomogeneous equation, using the Wronskian and either a system or direct integration to find the complementary and particular solutions.

Solve second-order nonhomogeneous equations by forming the associated homogeneous equation, building the complementary and particular solutions, and applying initial conditions to determine C1 and C2.

Explore how differential equations model real-world phenomena using direction fields to visualize solution curves and initial value problems. See examples such as population growth and the flow of electrical current.

Explore direction fields as the geometric graph of differential equations, where solution curves follow tangents and reveal the general solution family.

Define the interval of validity as the domain where a particular solution exists. For linear equations, this is all real numbers; for nonlinear cases, find where the solution becomes undefined.

Explore Euler's method, a numerical iterative approach that updates y by y_n = y_{n-1} + f(t_{n-1}, y_{n-1}) delta t to approximate differential equations.

Explore autonomous differential equations and equilibrium solutions; analyze stability via direction fields, highlighting stable and unstable steady states and their long-term behavior.

The logistic equation models population growth constrained by the environment. It shows slow growth at low numbers, fast growth mid-range, and equilibria at zero and carrying capacity.

Study predator-prey dynamics with two-population differential equations, distinguishing cooperative and competitive interactions by positive or negative terms. Identify equilibria, zero-zero, zero-a, a-zero, and the balanced (a,b).

Apply the differential equation dP/dt = kP to derive P(t) = P0 e^{kt} and model exponential growth or decay with growth or decay constants using initial conditions.

Model mixing problems with first-order differential equations, using input and output rates to track salt in a tank's brine, solving for y(t) with initial conditions.

Learn Newton's law of cooling, a separable differential equation for temperature decay toward ambient. Use k and initial conditions to compute when a cooling object reaches a target temperature.

Model second-order differential equations for circuit current in a series L-R-C using Kirchhoff's second law to derive L q'' + R q' + (1/C) q = E(t) and analyze damping.

Explore how spring-mass systems are modeled by second-order differential equations, covering free and damped motion, Hooke's law, equilibrium, and driven motion.

Explore expressing solutions to first order and second order differential equations as power series, review power series basics, and learn how to find power series solutions.

Explore power series basics, including centered forms, convergence radii, and ratio test, then apply Maclaurin series and differentiation to build series solutions for differential equations.

Master adding power series by ensuring matching indices and in phase, fix misaligned starts with index shifts and substitutions, then express the sum as a power series for differential equations.

Explore power series solutions for second-order linear homogeneous differential equations with polynomial coefficients, identify ordinary points, derive recurrence relations, and verify with y'-x y=0 yielding e^{x^2/2}.

Explore solving differential equations with nonpolynomial coefficients by using power series, substituting Maclaurin series like sine x, and deriving the coefficient relations for the general solution.

Examine singular points and Frobenius' theorem to obtain power series solutions around a singular point. Classify regular versus irregular, and derive y=(x-x0)^r with recurrence relations.

Explore Laplace transforms as an operator that converts differential equations into algebraic ones, and learn the inverse transform, plus applications to step functions, the direct delta function, and convolution integrals.

Learn Laplace transform as a method for solving differential equations with discontinuous forcing, transforming f(t) to F(s) via ∫0∞ e^{-st} f(t) dt; e^{-2t} maps to 1/(s+2) for s>-2.

Leverage a table of Laplace transforms to quickly compute transforms by identifying patterns such as e^{at}, t^n, sine and cosine, and using linearity and initial conditions.

Explore when Laplace transforms exist by applying piecewise continuity and exponential type, and compare exponential order with examples like e^{-2t}, cosine t, and e^{t^2}.

Master partial fractions decompositions by factoring the denominator completely, handling distinct and repeated linear and quadratic factors, and solving for constants using the cover up method or coefficient equations.

Use inverse Laplace transforms to convert capital F of S back to lowercase f of t, often via partial fractions and the transform table.

Derive the Laplace transform formulas for the first and second derivatives using integration by parts, obtaining L{f'}=sF(s)−f(0) and L{f''}=s^2F(s)−sf(0)−f'(0), then apply initial conditions y(0)=1, y'(0)=0 to get L{y'}=sY(s)−1 and L{y''}=s^2Y(s)−s.

Apply Laplace transforms to solve initial value problems for second-order differential equations, perform transforms and shifting for nonzero initial times, then use partial fractions and inverse transforms to get solution.

learn how step functions, including the unit step and heaviside function, switch on at t=c and can take nonconstant values; build piecewise signals with shifts, scales, and one-minus forms.

The second shifting theorem uses a unit step to shift a function by c, turning it off before c and yields the Laplace transform e^{-cs} times the original F(s).

Extend the second shifting theorem to Laplace transforms of step functions, including the step transform e^-Cs/s, and learn to fix mismatched shifts by shifting g(t) left and applying partial fractions.

Learn to solve initial value problems with step-function forcing by checking shifts, applying Laplace transforms, and using partial fractions and inverse transforms to obtain y(t).

Explore the Dirac delta function and its unit impulse properties, then apply the Laplace transform to solve a delta-driven initial-value problem.

Explore how the Laplace transform of a product uses convolution, not direct multiplication, by forming F(t)*G(t) and taking its transform. An example with F(t)=t and g(t)=e^t illustrates the method.

Learn how convolution integrals solve initial value problems with a generic forcing function g(t) using the Laplace transform, yielding a general solution via f(t) and g(t) through convolution.

Learn to solve systems of differential equations using matrices and vector equations. Solve homogeneous and nonhomogeneous systems for the general solution.

Master matrix basics: dimensions, rectangular and square forms, and representing systems of equations in matrix form. Explore multiplication, determinants, the identity matrix, and Gauss-Jordan elimination.

Build a system of differential equations from higher-order forms, express it as x' = A x + F, and distinguish homogeneous from non homogeneous cases.

Learn to solve systems of differential equations via a matrix form, building independent solution vectors and applying complementary and particular solutions for homogeneous and nonhomogeneous cases.

Derive eigenvalue eigenvector pairs from the characteristic equation det(A−λI)=0 for a 2×2 system, form the general solution with X1 e^(λ1 t) and X2 e^(λ2 t), and apply initial conditions.

Learn to solve systems with equal real eigenvalues of multiplicity two, distinguish cases with multiple eigenvectors versus a single eigenvector, and derive the general solution using x1 and x2.

Explore how to handle equal eigenvalues with multiplicity three by comparing algebraic and geometric multiplicities, recognizing no defect or defect cases, and forming the three solution vectors x1, x2, x3.

Explore solving systems of differential equations with complex conjugate eigenvalues, determine lambda = alpha ± beta i, and convert complex solutions into real terms using cosine and sine.

Explore phase portraits of homogeneous systems with distinct real eigenvalues, identifying unstable propeller nodes, stable attractor nodes, and saddles via eigenvectors, directions, and the zero vector equilibrium baseline.

Explore phase portraits for equal real eigenvalues, including non singular and singular matrices, one or two independent eigenvectors, and node or singular node behavior along lines of equilibrium.

Explore phase portraits for complex eigenvalues, identifying centers (zero real part), unstable repelling spirals (positive real part), and attractor spirals (negative real part) using the one-zero test to determine direction.

Solve nonhomogeneous systems by splitting into complementary and particular solutions. Use undetermined coefficients or variation of parameters, with polynomial, exponential, and sine-cosine guesses, for two- and three-equation cases.

Apply variation of parameters to solve nonhomogeneous linear systems by constructing the complementary solution, forming the fundamental matrix phi(t), and obtaining a particular solution via the integral of phi(t)^{-1} f(t).

Learn how the matrix exponential solves non-homogeneous systems by giving the general solution as the sum of complementary and particular solutions, via power series or inverse Laplace transform.

Explore solving third to higher order differential equations, starting with homogeneous cases and applying undetermined coefficients, variation of parameters, Laplace transforms, and power theory solutions for nonhomogeneous higher order equations.

Explore solving homogeneous higher-order differential equations, put in standard form, and construct general solutions from linearly independent roots. Learn distinct real, equal real, and complex conjugate roots and their multiplicities.

Extend undetermined coefficients to higher-order nonhomogeneous equations by solving the associated homogeneous equation for the complementary solution, then construct a particular solution, adjusting for overlaps due to repeated roots.

Extend variation of parameters to higher order differential equations using a system of equations and Cramer's rule, compute Wronskian determinants, and obtain the particular and general solutions.

Learn how to apply Laplace transforms to higher order differential equations, deriving general solutions by transforming derivatives, applying initial conditions, and using patterns for terms up to the fifth derivative.

Extend the two-by-two system approach to three-by-three systems, classifying eigenvalues as real, complex conjugates, or repeated, compute eigenvectors, and form three independent solutions for the general solution.

Use power series to solve differential equations. Align indices and derive a recurrence to form the general solution with C0, C1, C2, y''' + y' - x y = 0.

Learn to represent a function as a Fourier series by decomposing it into sums of sine and cosine terms, including cosine-only and sine-only series, and apply to piecewise functions.

Explore Fourier series representations as an alternative to Taylor series, derive a0, an, bn, and use even/odd properties, integration by parts, and periodicity to express functions as sine-cosine sums.

Periodic functions repeat values at intervals, and periodic extensions turn nonperiodic functions into periodic ones to enable Fourier series, with even and odd extensions on the interval -L to L.

Compute the 40 series representation of a piecewise function by splitting the integrals for a0, an, and bn across -L to 0 and 0 to L.

Master the convergence of Fourier series for piecewise smooth functions and their periodic extensions, including convergence at continuous points and the average of one-sided limits at jump discontinuities.

Explore how the Fourier cosine series represents even functions with only cosines, while non-even functions use even extension on 0 to L to form a cosine-only series.

Learn to derive the Fourier sine series for odd and non-odd functions via odd extensions on 0 to L, using integration by parts to compute B_n.

Learn to derive cosine and sine series for piecewise functions by using even and odd extensions. Compute the corresponding Fourier coefficients a0, an, and bn and assemble the series.

Introduce partial differential equations by reviewing partial derivatives and distinguishing boundary value problems from initial value problems. Explore the heat equation and the pluses equation.

Explore the separation of variables for partial differential equations, using a product solution to convert a partial differential equation into two ordinary differential equations with a separation constant, lambda.

Explore boundary value problems for second order differential equations, using separation of variables and a product solution to translate PDEs to ODEs with boundary conditions like u(0,t)=0 and u(L,t)=0.

Solve the heat equation by separation of variables, turning it into two ordinary differential equations with boundary conditions, yielding a sine series in x and exponential decay in t.

Set nonzero boundary temperatures in the one-dimensional heat equation, determine the equilibrium temperature u_e(x), and solve with u(x,t)=u_e(x)+v(x,t) via a Fourier sine series.

Explore Laplace's equation as a two-variable partial differential equation, solving a rectangular boundary value problem with separation of variables, product solutions, and Fourier-like boundary series.