[{"data":1,"prerenderedAt":1200},["ShallowReactive",2],{"subject:differential-equations":3,"course-wordcounts":75,"nav:differential-equations":987},{"id":4,"title":5,"blurb":6,"body":7,"brief":18,"category":56,"description":57,"draft":58,"extension":59,"meta":60,"module":15,"navigation":25,"path":61,"practice":62,"rawbody":63,"readingTime":64,"seo":69,"sources":70,"status":71,"stem":72,"summary":15,"topics":73,"__hash__":74},"course\u002F03.differential-equations\u002Findex.md","Differential Equations","Equations whose unknowns are functions — solving them exactly when\npossible, qualitatively when not, and numerically when all else fails; from\nfirst-order models to chaos and Fourier series.\n",{"type":8,"value":9,"toc":14},"minimark",[10],[11,12,13],"p",{},"A differential equation is a law written in the language of change: it fixes a\nfunction by constraining its derivatives rather than its values. These notes\nbegin with what that means geometrically — direction fields and the solution\ncurves that thread them — then build the exact machinery for the equations that\nyield to it: separable and first-order linear equations, the second-order linear\ntheory behind every oscillator, and the series and Laplace-transform techniques\nthat handle the rest. From there the subject turns to systems and their phase\nportraits, where eigenvalues classify the flow into nodes, saddles, and spirals,\nand then to the nonlinear world of equilibria, stability, limit cycles, and\nchaos, where qualitative reasoning replaces closed forms. Numerical integrators\nsupply answers when nothing else will, and Fourier series and transforms carry\nthe methods over to the classical partial differential equations of heat, waves,\nand potential. Boyce & DiPrima is the spine; Simmons supplies the history and the\napplications that motivate each method.",{"title":15,"searchDepth":16,"depth":16,"links":17},"",2,[],[19,21,26,28,32,34,36,40,42,46,48,52,54],{"p":20},"A differential equation relates a function to its own rates of change.\nYou rarely get the function handed to you; instead you get a law it must\nobey at every instant, and the work is recovering the function from that\nlaw.\n",{"fig":22,"n":23,"caption":24,"large":25},"de-slope-field","001","A direction field for y' = y(1−y), with one solution curve threaded so\nits tangent matches the field everywhere.\n",true,{"p":27},"The first thing to read off an equation is its \u003Cstrong>geometry\u003C\u002Fstrong>.\nA first-order law assigns a slope to every point of the plane, and its\nsolutions trace the curves that stay tangent to that field everywhere, so\nyou can see their shape long before you can write a formula.\n",{"fig":29,"n":30,"caption":31},"de-phase-portrait","002","A phase portrait: a linear system with complex eigenvalues spirals into a\nstable equilibrium.\n",{"p":33},"For linear systems the geometry becomes a \u003Cstrong>phase portrait\u003C\u002Fstrong>.\nThe eigenvalues of the coefficient matrix decide everything — real and\nnegative gives decay, imaginary gives rotation, positive real part sends\ntrajectories away — and the whole flow sorts into nodes, saddles, and\nspirals.\n",{"p":35},"Exact methods come next: separation, integrating factors, and\ncharacteristic equations solve the first- and second-order equations\noutright, while series and the Laplace transform reach the ones that\nresist elementary functions.\n",{"fig":37,"n":38,"caption":39},"de-oscillator","003","A damped oscillation e^{−γt}cos ωt, decaying inside its exponential\nenvelope.\n",{"p":41},"The second-order linear equation is the workhorse of physics: a mass on a\nspring, a circuit, a swinging pendulum. With damping its solution is an\noscillation trapped inside a shrinking exponential envelope — the balance\nof restoring force and dissipation made visible.\n",{"fig":43,"n":44,"caption":45},"de-exponential","004","y' = k y: pure exponential growth for k>0, decay for k\u003C0, from one shared\nseed.\n",{"p":47},"Underneath all of it sits the simplest equation, y' = k y, whose solution\nis a bare exponential. Growth, decay, cooling, and half-lives are all this\none law with a sign chosen; every richer model is measured against it.\n",{"fig":49,"n":50,"caption":51},"de-limit-cycle","005","A predator-prey limit cycle: an isolated closed orbit that nearby\ntrajectories spiral onto.\n",{"p":53},"Most equations that matter are \u003Cem>nonlinear\u003C\u002Fem>, and there the goal\nshifts from formulas to behaviour — equilibria, stability, limit cycles,\nand the onset of chaos, questions you answer qualitatively even when no\nclosed form exists.\n",{"p":55},"When even that fails you integrate numerically, stepping the solution\nforward in small increments; and Fourier methods close the loop, turning\nthe classical partial differential equations of heat, waves, and\npotential back into ordinary ones.\n","math","A differential equation relates a function to its rates of change, and most\nof physics, engineering, and population dynamics is written in that\nlanguage. These notes cover exact solution methods for first- and\nsecond-order equations, series and Laplace-transform techniques, linear\nsystems and phase portraits, stability and chaos, numerical integrators,\nand the Fourier methods that solve the classical partial differential\nequations. Boyce & DiPrima is the spine; Simmons supplies history and\napplications.\n",false,"md",{},"\u002Fdifferential-equations",[],"---\ntitle: Differential Equations\nstatus: available\ncategory: math\nblurb: |\n  Equations whose unknowns are functions — solving them exactly when\n  possible, qualitatively when not, and numerically when all else fails; from\n  first-order models to chaos and Fourier series.\ndescription: |\n  A differential equation relates a function to its rates of change, and most\n  of physics, engineering, and population dynamics is written in that\n  language. These notes cover exact solution methods for first- and\n  second-order equations, series and Laplace-transform techniques, linear\n  systems and phase portraits, stability and chaos, numerical integrators,\n  and the Fourier methods that solve the classical partial differential\n  equations. Boyce & DiPrima is the spine; Simmons supplies history and\n  applications.\nbrief:\n  - p: |\n      A differential equation relates a function to its own rates of change.\n      You rarely get the function handed to you; instead you get a law it must\n      obey at every instant, and the work is recovering the function from that\n      law.\n  - fig: de-slope-field\n    n: \"001\"\n    caption: |\n      A direction field for y' = y(1−y), with one solution curve threaded so\n      its tangent matches the field everywhere.\n    large: true\n  - p: |\n      The first thing to read off an equation is its \u003Cstrong>geometry\u003C\u002Fstrong>.\n      A first-order law assigns a slope to every point of the plane, and its\n      solutions trace the curves that stay tangent to that field everywhere, so\n      you can see their shape long before you can write a formula.\n  - fig: de-phase-portrait\n    n: \"002\"\n    caption: |\n      A phase portrait: a linear system with complex eigenvalues spirals into a\n      stable equilibrium.\n  - p: |\n      For linear systems the geometry becomes a \u003Cstrong>phase portrait\u003C\u002Fstrong>.\n      The eigenvalues of the coefficient matrix decide everything — real and\n      negative gives decay, imaginary gives rotation, positive real part sends\n      trajectories away — and the whole flow sorts into nodes, saddles, and\n      spirals.\n  - p: |\n      Exact methods come next: separation, integrating factors, and\n      characteristic equations solve the first- and second-order equations\n      outright, while series and the Laplace transform reach the ones that\n      resist elementary functions.\n  - fig: de-oscillator\n    n: \"003\"\n    caption: |\n      A damped oscillation e^{−γt}cos ωt, decaying inside its exponential\n      envelope.\n  - p: |\n      The second-order linear equation is the workhorse of physics: a mass on a\n      spring, a circuit, a swinging pendulum. With damping its solution is an\n      oscillation trapped inside a shrinking exponential envelope — the balance\n      of restoring force and dissipation made visible.\n  - fig: de-exponential\n    n: \"004\"\n    caption: |\n      y' = k y: pure exponential growth for k>0, decay for k\u003C0, from one shared\n      seed.\n  - p: |\n      Underneath all of it sits the simplest equation, y' = k y, whose solution\n      is a bare exponential. Growth, decay, cooling, and half-lives are all this\n      one law with a sign chosen; every richer model is measured against it.\n  - fig: de-limit-cycle\n    n: \"005\"\n    caption: |\n      A predator-prey limit cycle: an isolated closed orbit that nearby\n      trajectories spiral onto.\n  - p: |\n      Most equations that matter are \u003Cem>nonlinear\u003C\u002Fem>, and there the goal\n      shifts from formulas to behaviour — equilibria, stability, limit cycles,\n      and the onset of chaos, questions you answer qualitatively even when no\n      closed form exists.\n  - p: |\n      When even that fails you integrate numerically, stepping the solution\n      forward in small increments; and Fourier methods close the loop, turning\n      the classical partial differential equations of heat, waves, and\n      potential back into ordinary ones.\n---\n\nA differential equation is a law written in the language of change: it fixes a\nfunction by constraining its derivatives rather than its values. These notes\nbegin with what that means geometrically — direction fields and the solution\ncurves that thread them — then build the exact machinery for the equations that\nyield to it: separable and first-order linear equations, the second-order linear\ntheory behind every oscillator, and the series and Laplace-transform techniques\nthat handle the rest. From there the subject turns to systems and their phase\nportraits, where eigenvalues classify the flow into nodes, saddles, and spirals,\nand then to the nonlinear world of equilibria, stability, limit cycles, and\nchaos, where qualitative reasoning replaces closed forms. Numerical integrators\nsupply answers when nothing else will, and Fourier series and transforms carry\nthe methods over to the classical partial differential equations of heat, waves,\nand potential. Boyce & DiPrima is the spine; Simmons supplies the history and the\napplications that motivate each method.\n\n",{"text":65,"minutes":66,"time":67,"words":68},"1 min 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Direction Fields, and Solution Curves","\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields",[989],"A differential equation relates an unknown function to its own rates of change. Three first-order models — a falling body, a cooling object, a population under predation — share the form dy\u002Fdt = ay - b; the slope field fixes their equilibria and long-run behavior before any formula is found. Solving the linear case gives the general solution, its integral curves, and the particular solution selected by an initial condition.\n",{"title":999,"path":1000,"lessonNumber":16,"topics":1001,"summary":1002},"Classifying Equations: Order, Linearity, ODE vs. PDE","\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology",[989],"Every solution method targets a specific class of equation, so the first question about any differential equation is which classes it belongs to. Four independent axes sort them: ordinary versus partial, order, linear versus nonlinear, and homogeneous versus nonhomogeneous. Systems, verification of a solution by substitution, and the split between initial and boundary value problems complete the vocabulary.\n",{"module":1004,"moduleNumber":16,"slug":1005,"lessons":1006},"First-Order Equations","first-order",[1007,1012,1017,1023,1029,1035],{"title":1008,"path":1009,"lessonNumber":990,"topics":1010,"summary":1011},"Linear Equations and Integrating Factors","\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors",[1004],"A first-order linear equation has the unknown and its derivative to the first power only. Multiplying by an integrating factor collapses the left side into a single derivative, and one integration gives the general solution in closed form. The solution exists wherever the coefficients are continuous, and for a constant coefficient it splits into a decaying transient and a steady state set by the forcing.\n",{"title":1013,"path":1014,"lessonNumber":16,"topics":1015,"summary":1016},"Separable and Exact Equations","\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact",[1004],"Two nonlinear first-order classes solve by direct integration. A separable equation splits so that each variable can be integrated on its own side, giving an implicit relation. An exact equation is the total differential of a hidden potential function, recognized by a symmetry test on its coefficients; when the test fails, an integrating factor can sometimes restore exactness. A change of variable brings homogeneous equations into the separable class.\n",{"title":1018,"path":1019,"lessonNumber":1020,"topics":1021,"summary":1022},"Modeling with First-Order Equations","\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order",3,[1004],"A rate law is a differential equation. Each first-order model starts from one governing principle: conservation of mass for a mixing tank, proportional change for interest and radioactive decay, Newton's law of cooling, a force balance for a body falling against drag, and Kirchhoff's law for a series circuit. Setting the derivative to zero recovers the steady state, and the transient records how the initial condition relaxes toward it.\n",{"title":1024,"path":1025,"lessonNumber":1026,"topics":1027,"summary":1028},"Autonomous Equations, Phase Lines, and Population Dynamics","\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics",4,[1004],"An autonomous equation y' = f(y) can be analyzed qualitatively without being solved. Its constant solutions are the zeros of f, and the sign of f between them fixes whether nearby solutions rise or fall, which the phase line records as a column of arrows. The logistic and threshold models, constant- and effort-proportional harvesting, and the properties nonlinear equations lose all follow from this reading.\n",{"title":1030,"path":1031,"lessonNumber":1032,"topics":1033,"summary":1034},"Existence, Uniqueness, and Euler's Method","\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler",5,[1004],"Existence and uniqueness can be settled before any attempt to solve. The existence-uniqueness theorem gives sufficient conditions on f, and a standard example shows what fails when they do not hold. Picard's successive approximations build the solution as the limit of an iteration, and Euler's method turns the same tangent-line idea into a numerical procedure for the equations no formula reaches.\n",{"title":1036,"path":1037,"lessonNumber":1038,"topics":1039,"summary":1040},"First-Order Difference Equations","\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations",6,[1004],"A difference equation advances a sequence one index at a time by a rule y_{n+1} = f(y_n). The linear case y_{n+1} = rho*y_n + b solves in closed form and converges to its equilibrium exactly when the ratio has magnitude below one, which underlies compound-interest and loan calculations. The logistic difference equation shows the nonlinear counterpart: an exchange of stability, a cascade of period doublings, and the onset of chaos.\n",{"module":1042,"moduleNumber":1020,"slug":1043,"lessons":1044},"Second-Order Linear Equations","second-order-linear",[1045,1050,1055,1060,1065,1070],{"title":1046,"path":1047,"lessonNumber":990,"topics":1048,"summary":1049},"Homogeneous Equations, the Wronskian, and Real Roots","\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients",[1042],"A second-order linear homogeneous equation with constant coefficients is solved by guessing an exponential and reducing to the quadratic characteristic equation. Two solutions span every solution exactly when their Wronskian is nonzero; that condition, superposition, and Abel's formula give the full structure of the general solution for the case of two distinct real roots.\n",{"title":1051,"path":1052,"lessonNumber":16,"topics":1053,"summary":1054},"Complex Roots, Repeated Roots, and Reduction of Order","\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots",[1042],"When the characteristic equation has complex conjugate roots, Euler's formula converts the complex exponentials into a real fundamental set of decaying or growing oscillations. When it has a repeated root, one exponential is lost and reduction of order recovers the missing second solution as $t\\,e^{rt}$. The same substitution $y = v(t)y_1(t)$ finds a second solution from any known one.\n",{"title":1056,"path":1057,"lessonNumber":1020,"topics":1058,"summary":1059},"Nonhomogeneous Equations: Undetermined Coefficients","\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients",[1042],"The general solution of a nonhomogeneous linear equation is a complementary solution plus any one particular solution. When the forcing term is a polynomial, exponential, sine, or cosine, a particular solution can be found by assuming a trial form of the same shape with unknown coefficients and solving for them. The one complication is resonance, handled by multiplying the trial by a power of $t$.\n",{"title":1061,"path":1062,"lessonNumber":1026,"topics":1063,"summary":1064},"Variation of Parameters","\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters",[1042],"Variation of parameters finds a particular solution of any nonhomogeneous linear equation from a fundamental set of the homogeneous one. Replacing the constants in the complementary solution by functions and imposing one convenient constraint reduces the problem to a two-by-two linear system whose solution is expressed through the Wronskian, giving an integral formula that works for forcing terms undetermined coefficients cannot touch.\n",{"title":1066,"path":1067,"lessonNumber":1032,"topics":1068,"summary":1069},"Mechanical and Electrical Vibrations","\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations",[1042],"A spring-mass-damper obeys a second-order linear equation, and so does a series RLC circuit, with the same mathematics governing both. Free undamped motion is a pure sinusoid; damping adds a decaying envelope with three regimes; periodic forcing produces a transient that dies out and a steady-state oscillation whose amplitude peaks sharply near the natural frequency, the phenomenon of resonance.\n",{"title":1071,"path":1072,"lessonNumber":1038,"topics":1073,"summary":1074},"Higher-Order Linear Equations","\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear",[1042],"The second-order theory extends directly to order $n$: the solution space is $n$-dimensional, spanned by any $n$ solutions with nonzero Wronskian. For constant coefficients the characteristic polynomial has degree $n$, and its roots (counted with multiplicity, real and complex) build the basis by the same rules as before. Coupled oscillators are the natural application that raises the order.\n",{"module":1076,"moduleNumber":1026,"slug":1077,"lessons":1078},"Series Solutions and Special Functions","series-solutions",[1079,1084,1089],{"title":1080,"path":1081,"lessonNumber":990,"topics":1082,"summary":1083},"Power Series Solutions Near Ordinary Points","\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points",[1076],"A linear equation with variable coefficients has no characteristic equation. A power series substituted into the equation matches coefficients to a recurrence relation, which near an ordinary point yields two independent analytic solutions. The radius of convergence is at least the distance from the expansion point to the nearest singular point in the complex plane.\n",{"title":1085,"path":1086,"lessonNumber":16,"topics":1087,"summary":1088},"Euler Equations, Regular Singular Points, and Frobenius","\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius",[1076],"The Euler equation x^2 y'' + a x y' + b y = 0 is solved outright by y = x^r, and its three root cases fix the behavior at any regular singular point. The Frobenius method multiplies x^r by a power series; the indicial equation chooses the exponents, and equal or integer-separated roots force a logarithm in the second solution. Gauss's hypergeometric equation is the archetype containing most classical functions as special cases.\n",{"title":1090,"path":1091,"lessonNumber":1020,"topics":1092,"summary":1093},"Bessel's Equation, Legendre Polynomials, and Special Functions","\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions",[1076],"Bessel's equation puts the Frobenius machinery through all three of its cases and produces the functions J and Y that govern anything vibrating or diffusing with circular symmetry. The gamma function extends the factorial so that Bessel functions of every order make sense; Legendre's equation, run through the hypergeometric form, yields the polynomials that play the same role in spherical geometry. Orthogonality ties both families to the eigenfunction expansions of Sturm–Liouville theory.\n",{"module":1095,"moduleNumber":1032,"slug":1096,"lessons":1097},"The Laplace Transform","laplace",[1098,1103],{"title":1099,"path":1100,"lessonNumber":990,"topics":1101,"summary":1102},"The Laplace Transform: Definition, Properties, and Solving IVPs","\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps",[1095],"The Laplace transform sends a function of time to a function of a complex frequency by integrating it against the kernel e^{-st}. Differentiation in t becomes multiplication by s, so a linear constant-coefficient initial value problem turns into an algebraic equation. Existence rests on piecewise continuity and exponential order; the derivative rule folds in the initial data; and inversion runs through a transform table and partial fractions.\n",{"title":1104,"path":1105,"lessonNumber":16,"topics":1106,"summary":1107},"Step Functions, Discontinuous Forcing, Impulses, and Convolution","\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution",[1095],"The Heaviside step function and the second shifting theorem transform switches and discontinuous forcing into exponential factors on the transform. The Dirac delta idealizes an instantaneous impulse and transforms to a pure exponential. The convolution theorem inverts a product of transforms, writes the forced response as the impulse response convolved with the input, and solves Abel's tautochrone by transform.\n",{"module":1109,"moduleNumber":1038,"slug":1110,"lessons":1111},"Systems of First-Order Linear Equations","systems",[1112,1117,1122],{"title":1113,"path":1114,"lessonNumber":990,"topics":1115,"summary":1116},"Matrices, Linear Systems, and the Eigenvalue Toolkit","\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review",[1109],"Any nth-order linear equation, and any coupled collection of them, rewrites as a single first-order system x' = P(t)x + g(t). The matrix and vector algebra behind that form, the eigenvalue problem det(A - λI) = 0 that drives every solution method, and the fundamental theory — superposition, the Wronskian, Abel's theorem — together establish that n independent solutions span all solutions.\n",{"title":1118,"path":1119,"lessonNumber":16,"topics":1120,"summary":1121},"Homogeneous Constant-Coefficient Systems and Phase Portraits","\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits",[1109],"For x' = Ax with A constant, the trial x = ξe^{rt} turns the differential equation into the eigenvalue problem Aξ = rξ. The eigenvalues fix the geometry of the phase plane: real opposite signs give a saddle, real same sign a node, complex a spiral, purely imaginary a center. Worked in the plane, these cases form the eigenvalue-type classification of equilibria.\n",{"title":1123,"path":1124,"lessonNumber":1020,"topics":1125,"summary":1126},"Repeated Eigenvalues, Fundamental Matrices, and Nonhomogeneous Systems","\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices",[1109],"When a repeated eigenvalue supplies too few eigenvectors, a generalized eigenvector supplies the missing solution as ξte^{ρt} + ηe^{ρt}, giving an improper node. A fundamental set packaged as a matrix Φ(t) yields the matrix exponential e^{At}, the propagator mapping initial states to later ones. Variation of parameters solves the nonhomogeneous system x' = Ax + g(t).\n",{"module":1128,"moduleNumber":1129,"slug":1130,"lessons":1131},"Numerical Methods",7,"numerical",[1132,1137],{"title":1133,"path":1134,"lessonNumber":990,"topics":1135,"summary":1136},"Euler, Improved Euler, and Runge–Kutta","\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta",[1128],"Most initial value problems have no closed-form solution, so the solution is approximated on a grid. Euler's method steps along the tangent line, the improved Euler method averages two slopes, and the classical Runge–Kutta method averages four. Each added stage raises the order of accuracy at the cost of more evaluations per step, measured by how the local and global truncation errors scale with the step size.\n",{"title":1138,"path":1139,"lessonNumber":16,"topics":1140,"summary":1141},"Multistep Methods, Systems, and Stability","\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability",[1128],"One-step methods discard everything but the last point. Multistep methods fit a polynomial to several past values and integrate it forward: the explicit Adams–Bashforth formulas, the implicit and more accurate Adams–Moulton formulas, and predictor–corrector pairs that combine them. The same rules extend verbatim to systems in vector form. A separate concern is stability: round-off can dominate truncation, and stiff equations force a tiny step for stability even when accuracy would allow a large one.\n",{"module":1143,"moduleNumber":1144,"slug":1145,"lessons":1146},"Nonlinear Systems and Stability",8,"nonlinear",[1147,1152,1157],{"title":1148,"path":1149,"lessonNumber":990,"topics":1150,"summary":1151},"The Phase Plane, Critical Points, and Stability","\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability",[1143],"Most nonlinear systems cannot be solved in closed form, so they are studied geometrically. The phase plane turns an autonomous planar system into a family of trajectories; the five archetypes of critical point follow from the eigenvalues of the coefficient matrix; the trace-determinant plane reads off type and stability directly; and epsilon-delta definitions make stability, asymptotic stability, and instability precise.\n",{"title":1153,"path":1154,"lessonNumber":16,"topics":1155,"summary":1156},"Locally Linear Systems and Liapunov's Method","\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov",[1143],"Near a critical point a nonlinear system looks linear, and the linear part is the Jacobian. The linearization fixes the type and stability of the nonlinear critical point in every case except a center or a repeated eigenvalue. Liapunov's direct method settles those cases and bounds the basin of attraction by constructing an energy-like function, without solving the system.\n",{"title":1158,"path":1159,"lessonNumber":1020,"topics":1160,"summary":1161},"Population Models, Limit Cycles, and Chaos","\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles",[1143],"The phase-plane methods apply directly to interacting-population models. Competing species either coexist or drive one another to extinction, decided by a single inequality among the interaction constants; the Lotka-Volterra predator-prey system produces closed population cycles. Limit cycles and the Poincaré-Bendixson theorem, the van der Pol oscillator, and the Lorenz equations with their strange attractor carry the theory into chaos.\n",{"module":1163,"moduleNumber":1164,"slug":1165,"lessons":1166},"PDEs, Fourier Series, and Boundary Value Problems",9,"pdes-fourier-bvp",[1167,1175,1180],{"title":1168,"path":1169,"lessonNumber":990,"topics":1170,"summary":1174},"Fourier Series and Convergence","\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series",[1171,1172,1173],"PDEs","Fourier Series","and Boundary Value Problems","A two-point boundary value problem has nontrivial solutions only at a discrete set of eigenvalues, the same trichotomy that governs a singular linear system. For y'' + lambda y = 0 with zero endpoints the eigenfunctions are sines and cosines, and their orthogonality gives the Euler-Fourier coefficient formulas. The convergence theorem fixes when the series returns the function, the Gibbs phenomenon measures the overshoot at a jump, and even\u002Fodd symmetry produces half-range sine and cosine series.\n",{"title":1176,"path":1177,"lessonNumber":16,"topics":1178,"summary":1179},"Separation of Variables: Heat, Wave, and Laplace Equations","\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations",[1171,1172,1173],"Separation of variables replaces a partial differential equation by a pair of ordinary ones joined through a shared separation constant. Applied to the heat equation it produces the eigenvalue problem X'' + lambda X = 0, and the solution assembles as a Fourier series in the eigenfunctions. The same steps solve the wave equation, whose modes are standing waves, and Laplace's equation, the steady-state limit posed on a region rather than an interval.\n",{"title":1181,"path":1182,"lessonNumber":1020,"topics":1183,"summary":1184},"Sturm-Liouville Theory","\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville",[1171,1172,1173],"The eigenvalue problem behind separation of variables generalizes to the self-adjoint Sturm-Liouville form. Lagrange's identity makes the operator symmetric, and from that one fact follow real eigenvalues, orthogonal eigenfunctions, and eigenfunction expansions that behave like Fourier series. Singular problems admit Bessel and Legendre functions, and Sturm's separation and comparison theorems describe how the eigenfunctions oscillate.\n",{"module":1186,"moduleNumber":1187,"slug":1188,"lessons":1189},"Historical Notes and the Calculus of Variations",10,"history-variations",[1190,1195],{"title":1191,"path":1192,"lessonNumber":990,"topics":1193,"summary":1194},"The Calculus of Variations","\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations",[1186],"Ordinary calculus finds the point where a function is stationary; the calculus of variations finds the whole curve where an integral is stationary. Euler's differential equation is the necessary condition for an extremal, and it becomes integrable in three cases, solving the shortest-path, minimal-surface, and brachistochrone problems. Lagrange multipliers extend the method to isoperimetric constraints, and Hamilton's principle recovers Newton's law from a single stationary integral.\n",{"title":1196,"path":1197,"lessonNumber":16,"topics":1198,"summary":1199},"Great Problems and the People Who Solved Them","\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes",[1186],"Differential equations grew out of specific problems, not a plan: the invention of calculus by Newton and Leibniz, the Bernoulli brachistochrone challenge, Euler's flood of methods, Lagrange's analytical mechanics, Gauss and Riemann's rigor, Laplace's celestial mechanics, and Poincaré's qualitative theory. Each method descends from a named problem, and reading the subject forward from those problems explains why its parts fit together.\n",1786059477082]