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Every rule and concept in Differential Equations, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (105)

A checklist for a planar system: For $x' = f(x, y)$ , $y' = g(x, y)$ :; Nonlinear Systems & Linearization › Equilibria, Jacobians and Nullclines
A second solution from the first: If $y_1$ solves $y'' + p(x)y' + q(x)y = 0$ , a second, independent solution is \[ y_2 = y_1∫ $\frac{W}{y_1^2}$ dx, W = e^-∫ p dx.; The Wronskian & Variation of Parameters › Abel's Formula and Reduction of Order
An impulse kicks the velocity: For $y'' + by' + cy = k\,\delta(t - t_0)$ , the solution $y$ is continuous at $t_0$ and $y'$ jumps by exactly $k$ .; Laplace II: Steps, Impulses & Convolution › Impulses and the Impulse Response
Bessel functions: the drumhead cosine: Bessel's equation governs radial vibrations of circular membranes and heat in cylinders.; Frobenius & Special Functions › Series at Singular Points
Bifurcations: how equilibria are born and die: Add a parameter: $y' = r + y^2$ .; Existence, Uniqueness & Bifurcations › When Solutions Exist, Split, and Vanish
Boundary conditions quantize: An initial-value problem specifies everything at one point; a boundary-value problem splits conditions between two ( $y(0) = y(L) = 0$ ) — and suddenly solutions exist only for special parameter values.; Boundary Values & Fourier Series › From Boundary Conditions to Fourier
Complex eigenvalues spiral: Eigenvalues $\alpha \pm \beta i$ make trajectories rotate with rate $\beta$ while $e^{\alpha t}$ scales them: spirals in or out, or pure circles when $\alpha = 0$ .; Systems of Differential Equations › Linear Systems
Convolution: multiplication in disguise: The convolution $(f*g)(t) = \int_0^t f(\tau)g(t-\tau)\,d\tau$ satisfies $\mathcal L\{f*g\} = F(s)G(s)$ — products of transforms are convolutions of functions.; Laplace II: Steps, Impulses & Convolution › Discontinuous and Impulsive Forcing
Damping in one glance: For $y'' + by' + \omega^2y = 0$ , the discriminant $b^2 - 4\omega^2$ sorts the physics: negative = underdamped (oscillates while decaying), zero = critically damped (fastest settle, no overshoot), positive = overdamped.; Second-Order Linear Equations › Constant Coefficients
Derivatives become multiplication: $\mathcal L\{y'\} = s\,\mathcal L\{y\} - y(0)$ : differentiation turns into multiplying by $s$ , with the initial condition built in automatically.; Laplace Transforms › The Transform Method
Equations beyond elementary functions: The Airy-type equation $y'' = xy$ has no closed-form solution, but its recurrence $(n+2)(n+1)a_{n+2} = a_{n-1}$ generates the answer to any accuracy: with $y(0)=1$ , $y'(0)=0$ the series starts $1 + \tfrac{x^3}{6} + \cdots$ .; Series Solutions › The Power Series Method
Equilibria and stability: the phase-line shortcut: For an autonomous equation $y' = f(y)$ , enormous insight costs one sketch.; Models & Qualitative Behavior › Going Deeper: Explanations & Worked Problems
Exact equations: one function behind two coefficients: The form $M(x,y)\,dx + N(x,y)\,dy = 0$ is exact when some $F$ has $F_x = M$ and $F_y = N$ — then solutions are simply the level curves $F(x, y) = C$ .; Exact Equations & Substitutions › Exactness and the Great Substitutions
Follow the tangent, then re-aim: For $y' = f(t, y)$ , $y(t_0) = y_0$ and step $h$ : \[ t_n+1 = t_n + h, y_n+1 = y_n + h f(t_n, y_n).; Numerical Methods: Euler to Runge–Kutta › Euler's Method Step by Step
Fourier series: every shape from pure tones: The eigenfunctions are orthogonal ( $\int_0^\pi\sin mx\sin nx\,dx = 0$ for $m \ne n$ ), so any reasonable $f$ on $(0, \pi)$ expands as $f = \sum b_n\sin nx$ with each coefficient computed independently: $b_n = \tfrac{2}{\pi}\int_0^\pi f\sin nx\,dx$ — a projec…; Boundary Values & Fourier Series › From Boundary Conditions to Fourier
From eigenpairs to the phase portrait, step by step: To solve $\mathbf u' = A\mathbf u$ : find the eigenpairs of $A$ ; each gives a straight-line solution $e^{\lambda t}\mathbf v$ ; the general solution is $c_1e^{\lambda_1t}\mathbf v_1 + c_2e^{\lambda_2t}\mathbf v_2$ ; initial data fixes $c_1, c_2$ by writing $…; Systems of Differential Equations › Going Deeper: Explanations & Worked Problems
From Newton's law to the equation: A mass $m$ on a spring of stiffness $k$ , with a damper whose force is proportional to velocity (coefficient $c$ ), obeys \[ mx” + cx' + kx = 0, \] where $x(t)$ is the displacement from equilibrium.; Second-Order Linear Equations › The Spring–Mass System
Guess the shape of the forcing: For $y'' + by' + cy = f(x)$ : general solution $=$ homogeneous $+$ one particular solution.; Nonhomogeneous Equations & Resonance › Undetermined Coefficients
How many factors of x: If the exponent $k$ of the forcing $e^{kx}$ (or $\alpha + i\beta$ for $e^{\alpha x}\cos\beta x$ ) is a root of the characteristic equation, multiply the guess by $x^m$ , where $m$ is the multiplicity of that root: $m = 1$ for a simple root, $m = 2$ for a doub…; Nonhomogeneous Equations & Resonance › Resonance, Double Roots and Beats
Legendre polynomials: quantization by boundedness: Legendre's equation $(1-x^2)y'' - 2xy' + n(n+1)y = 0$ appears whenever a problem lives on a sphere.; Frobenius & Special Functions › Series at Singular Points
Linear: multiply by the integrating factor: For $y' + p(x)y = q(x)$ , multiply through by $\mu = e^{\int p\,dx}$ : the left side collapses to $(\mu y)'$ by design, and one integration finishes it.; First-Order Equations › The Two Workhorse Methods
Linearize at each equilibrium: A nonlinear system $x' = f(x,y)$ , $y' = g(x,y)$ is understood equilibrium by equilibrium: find all points where $f = g = 0$ , compute the jacobian $; Nonlinear Systems & Linearization › Local Pictures of Nonlinear Worlds
Logistic growth: $\tfrac{dP}{dt} = rP\left(1 - \tfrac{P}{K}\right)$ : growth is proportional to both the population and the remaining room.; Models & Qualitative Behavior › The Classic Models
Newton cooling & mixing tanks: Cooling: the excess over ambient decays exponentially — if the excess halves in 10 minutes it halves again in the next 10.; Models & Qualitative Behavior › The Classic Models
Picard's theorem: the rules of the game: If $f$ and $\partial f/\partial y$ are continuous near $(t_0, y_0)$ , the IVP $y' = f(t, y)$ , $y(t_0) = y_0$ has exactly one solution near $t_0$ .; Existence, Uniqueness & Bifurcations › When Solutions Exist, Split, and Vanish
Predator–prey and competition, read from Jacobians: Lotka–Volterra $x' = x - xy$ , $y' = -y + xy$ : at the coexistence point $(1,1)$ the Jacobian is $; Nonlinear Systems & Linearization › Local Pictures of Nonlinear Worlds
Proof idea: Picard iteration: Rewrite the IVP as the integral equation $y(t) = y_0 + \int_{t_0}^t f(s, y(s))\,ds$ and iterate: $y_{n+1}(t) = y_0 + \int_{t_0}^t f(s, y_n(s))\,ds$ starting from $y_0(t) \equiv y_0$ .; Existence, Uniqueness & Bifurcations › Proofs & Why It Matters
Proof sketch: Euler's error halves with h: One Euler step commits error $\approx \tfrac{h^2}{2}y''$ (the first Taylor term the tangent line misses).; Models & Qualitative Behavior › Proofs & Why It Matters
Proof: Abel's theorem: Differentiate $W = y_1y_2' - y_1'y_2$ : $W' = y_1y_2'' - y_1''y_2$ (the cross terms cancel).; The Wronskian & Variation of Parameters › Proofs & Why It Matters
Proof: coefficient matching is valid: A power series that sums to zero on an interval has all coefficients zero (differentiate repeatedly and evaluate at the center: $a_n = \tfrac{f^{(n)}(0)}{n!} = 0$ ).; Series Solutions › Proofs & Why It Matters
Proof: complex roots give cosine and sine (Euler): Euler's formula $e^{i\beta x} = \cos\beta x + i\sin\beta x$ (visible by comparing the three power series) turns the complex solutions $e^{(\alpha\pm i\beta)x}$ into $e^{\alpha x}(\cos\beta x \pm i\sin\beta x)$ .; Second-Order Linear Equations › Proofs & Why It Matters
Proof: e^λtv solves u′ = Au: Differentiate $\mathbf u(t) = e^{\lambda t}\mathbf v$ : $\mathbf u' = \lambda e^{\lambda t}\mathbf v$ .; Systems of Differential Equations › Proofs & Why It Matters
Proof: Euler's global error is O(h): One step from the true solution commits local error $y(t + h) - y(t) - hy'(t) = \tfrac{h^2}{2}y''(\xi)$ — the first Taylor term the tangent line drops.; Numerical Methods: Euler to Runge–Kutta › Proofs & Why It Matters
Proof: exponential solutions and half-life: $y' = ky$ separates to $\int\tfrac{dy}{y} = \int k\,dx$ , so $\ln|y| = kx + C$ and $y = y_0e^{kx}$ — and this is the only solution: if $z$ also solves it, then $(z e^{-kx})' = z'e^{-kx} - kze^{-kx} = 0$ , so $ze^{-kx}$ is constant.; First-Order Equations › Proofs & Why It Matters
Proof: general = homogeneous + particular: Let $y_p$ solve $Ly = f$ (writing $L$ for the whole left side).; Nonhomogeneous Equations & Resonance › Proofs & Why It Matters
Proof: Heun's extra order: Heun's update is $y + \tfrac h2\left[f(y) + f(y + hf(y))\right]$ .; Numerical Methods: Euler to Runge–Kutta › Proofs & Why It Matters
Proof: J₀ solves Bessel of order zero: Termwise, with $y = \sum c_kx^{2k}$ , $c_k = \tfrac{(-1)^k}{4^k(k!)^2}$ : compute $xy'' + y' + xy$ (the $\nu = 0$ equation divided by $x$ ).; Frobenius & Special Functions › Proofs & Why It Matters
Proof: Le^at = 1/(s−a): Directly: $\int_0^\infty e^{at}e^{-st}dt = \int_0^\infty e^{-(s-a)t}dt = \left[\dfrac{e^{-(s-a)t}}{-(s-a)}\right]_0^\infty = \dfrac{1}{s-a}$ for $s > a$ (the boundary term at $\infty$ dies because $s - a > 0$ ).; Laplace Transforms › Proofs & Why It Matters
Proof: Ltⁿ = n!/s^n+1: Induction with parts: $\mathcal L\{t^n\} = \int_0^\infty t^ne^{-st}dt = \tfrac ns\int_0^\infty t^{n-1}e^{-st}dt = \tfrac ns\,\mathcal L\{t^{n-1}\}$ (boundary terms vanish).; Laplace Transforms › Proofs & Why It Matters
Proof: Ly′ = sLy − y(0): Integrate by parts: $\int_0^\infty y'(t)e^{-st}dt = \left[y(t)e^{-st}\right]_0^\infty + s\int_0^\infty y(t)e^{-st}dt = -y(0) + s\,\mathcal L\{y\}$ , assuming $y$ grows slower than $e^{st}$ so the boundary term at infinity vanishes.; Laplace Transforms › Proofs & Why It Matters
Proof: Lotka–Volterra orbits really close up: The quantity $E(x, y) = x - \ln x + y - \ln y$ is conserved along solutions of $x' = x - xy$ , $y' = -y + xy$ .; Nonlinear Systems & Linearization › Proofs & Why It Matters
Proof: Newton cooling is exponential in the excess: The law says $T' = -k(T - T_{\text{amb}})$ .; Models & Qualitative Behavior › Proofs & Why It Matters
Proof: orthogonality of the sine family: Product-to-sum: $\sin mx\sin nx = \tfrac12\left[\cos(m-n)x - \cos(m+n)x\right]$ .; Boundary Values & Fourier Series › Proofs & Why It Matters
Proof: separation of variables for heat: Try $u = X(x)T(t)$ in $u_t = ku_{xx}$ : $xt' = kX''T$ , so $\dfrac{T'}{kT} = \dfrac{X''}{X}$ .; Boundary Values & Fourier Series › Proofs & Why It Matters
Proof: separation of variables is legitimate: If $\tfrac{dy}{dx} = g(x)h(y)$ with $h(y) \ne 0$ , divide: $\tfrac{y'(x)}{h(y(x))} = g(x)$ .; First-Order Equations › Proofs & Why It Matters
Proof: solving the logistic equation: Separate $\tfrac{dP}{P(1 - P/K)} = r\,dt$ and split by partial fractions: $\tfrac{1}{P(1-P/K)} = \tfrac1P + \tfrac{1/K}{1 - P/K}$ .; Models & Qualitative Behavior › Proofs & Why It Matters
Proof: the Bernoulli substitution linearizes: From $v = y^{1-n}$ : $v' = (1-n)y^{-n}y'$ .; Exact Equations & Substitutions › Proofs & Why It Matters
Proof: the Cauchy–Euler indicial equation: Substitute $y = x^r$ into $x^2y'' + \alpha xy' + \beta y = 0$ : the derivatives give $x^2\cdot r(r-1)x^{r-2} + \alpha x\cdot rx^{r-1} + \beta x^r = \left[r(r-1) + \alpha r + \beta\right]x^r$ .; Frobenius & Special Functions › Proofs & Why It Matters
Proof: the convolution theorem: $F(s)G(s) = \int_0^\infty\!\!\int_0^\infty f(\tau)g(\sigma)e^{-s(\tau+\sigma)}\,d\sigma\,d\tau$ — a double integral over the quarter-plane.; Laplace II: Steps, Impulses & Convolution › Proofs & Why It Matters
Proof: the double root needs xe^rx: If $r$ is a double root, then $\lambda^2 + b\lambda + c = (\lambda - r)^2$ , and comparing coefficients gives $b = -2r$ and $c = r^2$ .; Second-Order Linear Equations › Proofs & Why It Matters
Proof: the exactness test: If $F$ exists with $F_x = M$ , $F_y = N$ and $F$ is twice continuously differentiable, then $M_y = F_{xy} = F_{yx} = N_x$ by Clairaut's theorem — necessity.; Exact Equations & Substitutions › Proofs & Why It Matters
Proof: the integrating factor works: With $\mu = e^{\int p\,dx}$ , note $\mu' = p\mu$ (chain rule).; First-Order Equations › Proofs & Why It Matters
Proof: the resonant particular solution: For $y'' + \omega^2y = \cos\omega x$ , try $y = kx\sin\omega x$ : $y' = k\sin\omega x + k\omega x\cos\omega x$ and $y'' = 2k\omega\cos\omega x - k\omega^2x\sin\omega x$ .; Nonhomogeneous Equations & Resonance › Proofs & Why It Matters
Proof: the saddle-node signature: At a bifurcation the equilibrium equation $f(y; r) = 0$ must have a double root — a simple root moves smoothly as $r$ varies (implicit function theorem) and cannot vanish.; Existence, Uniqueness & Bifurcations › Proofs & Why It Matters
Proof: the t-shift rule: Directly: $\mathcal L\{g(t-a)u(t-a)\} = \int_a^\infty g(t-a)e^{-st}dt$ (the step kills $t < a$ ).; Laplace II: Steps, Impulses & Convolution › Proofs & Why It Matters
Proof: the variation-of-parameters formulas: With $y_p = u_1y_1 + u_2y_2$ and side condition $u_1'y_1 + u_2'y_2 = 0$ : then $y_p' = u_1y_1' + u_2y_2'$ , and $y_p'' = u_1'y_1' + u_2'y_2' + u_1y_1'' + u_2y_2''$ .; The Wronskian & Variation of Parameters › Proofs & Why It Matters
Proof: why e^rx and the characteristic equation: Substitute $y = e^{rx}$ into $y'' + by' + cy = 0$ : $e^{rx}(r^2 + br + c) = 0$ , and since $e^{rx} \ne 0$ , the exponent must satisfy $r^2 + br + c = 0$ .; Second-Order Linear Equations › Proofs & Why It Matters
Proof: why the Jacobian is the right linearization: Write $\mathbf x = \mathbf x^* + \mathbf u$ for a small displacement from the equilibrium $\mathbf x^*$ .; Nonlinear Systems & Linearization › Proofs & Why It Matters
Proof: x′ = y, y′ = −x moves in circles: Let $E(t) = x(t)^2 + y(t)^2$ .; Systems of Differential Equations › Proofs & Why It Matters
Proof: y′ = y forces aₙ = 1/n!: Substitute: $\sum_{n\ge1} na_nx^{n-1} = \sum_{n\ge0} a_nx^n$ .; Series Solutions › Proofs & Why It Matters
Radius at least the distance to the nearest singular point: For $P(x)y'' + Q(x)y' + R(x)y = 0$ with polynomial coefficients, the singular points are the zeros of $P$ — complex zeros included.; Series Solutions › How Far a Series Reaches
Rate in, rate out: A first-order model is a sentence about rates: $\dfrac{d(\text{amount})}{dt} = (\text{rate in}) - (\text{rate out})$ .; Models & Qualitative Behavior › Building a Model from Rates
Reading a bifurcation diagram: Plot the equilibria of $y' = f(y; r)$ against the parameter $r$ , solid where stable ( $f_y < 0$ ) and dashed where unstable ( $f_y > 0$ ).; Existence, Uniqueness & Bifurcations › Bifurcation Diagrams
Reading an equation: which method, and why: Given $y' = (\text{something})$ , the triage is: can the right side be factored as $g(x)h(y)$ — a pure- $x$ piece times a pure- $y$ piece?; First-Order Equations › Going Deeper: Explanations & Worked Problems
Real solutions from a complex eigenpair: If $A$ has eigenvalue $\lambda = \alpha + \beta i$ ( $\beta \ne 0$ ) with eigenvector $\mathbf v = \mathbf a + i\mathbf b$ , the real and imaginary parts of $e^{\lambda t}\mathbf v$ , \[ u_1 = e^α t( aβ t - bβ t), \…; Systems of Differential Equations › Complex Eigenvalues: Spirals and Centers
Regular singular points and the indicial equation: Where the leading coefficient vanishes (like $x = 0$ in $x^2y'' + xy' + (x^2 - \nu^2)y = 0$ ), plain power series can fail.; Frobenius & Special Functions › Series at Singular Points
Resonance: multiply by x: When the forcing already solves the homogeneous equation, the naive guess collapses — multiply it by $x$ .; Nonhomogeneous Equations & Resonance › Undetermined Coefficients
Separable: sort, then integrate: If $\tfrac{dy}{dx} = g(x)h(y)$ , move all $y$ ’s left and $x$ ’s right: $\tfrac{dy}{h(y)} = g(x)\,dx$ , integrate both sides, and solve for $y$ .; First-Order Equations › The Two Workhorse Methods
Sines and cosines together: A function on $[-L, L]$ (continued with period $2L$ ) has the Fourier series \[ f(x) ∼ $\frac{a_0}{2}$ + ∑_n≥1(a_n $\frac{n\pi x}{L}$ + b_n $\frac{n\pi x}{L}$ ), \] \[ a_n = 1L∫_-L^L f(x) $\frac{n\pi x}{L}$ dx, b_n = …; Boundary Values & Fourier Series › Full Fourier Series on [−L, L]
Solve along eigenvectors: For $\mathbf u' = A\mathbf u$ , each eigenpair gives a straight-line solution $e^{\lambda t}\mathbf v$ , and the general solution mixes them.; Systems of Differential Equations › Linear Systems
Stability: the other failure mode: Accuracy is not the only concern.; Numerical Methods: Euler to Runge–Kutta › Computing Solutions Honestly
Standard form first, then the factor: Before computing $\mu$ , divide by the coefficient of $y'$ so that the equation reads $y' + p(x)y = q(x)$ .; First-Order Equations › Linear Equations with Variable Coefficients
Substitute, shift, match: Write $y = \sum a_nx^n$ , substitute into the equation, shift indices so all powers align, and set each coefficient to zero — a recurrence for the $a_n$ .; Series Solutions › The Power Series Method
The characteristic equation: For $y'' + by' + cy = 0$ , try $y = e^{rx}$ : $r^2 + br + c = 0$ .; Second-Order Linear Equations › Constant Coefficients
The classical RK4 step: \[ k_1 = f(t_n, y_n), k_2 = f (t_n + h2, y_n + h2k_1), k_3 = f (t_n + h2, y_n + h2k_2), k_4 = f(t_n + h, y_n + hk_3), \] \[ y_n+1 = y_n + h6 (k_1 + 2k_2 + 2k_3 + k_4).; Numerical Methods: Euler to Runge–Kutta › Runge–Kutta Methods in Practice
The construction, and why g depends on y alone: To solve an exact equation $M\,dx + N\,dy = 0$ :; Exact Equations & Substitutions › Building the Potential, with Initial Data
The derivative test for equilibria: For $y' = f(y)$ with $f(y^*) = 0$ :; Existence, Uniqueness & Bifurcations › Stability from the Derivative
The formula in one line: With $y_1, y_2$ a fundamental pair for $y'' + p(x)y' + q(x)y = 0$ and $W = y_1y_2' - y_1'y_2$ , \[ y_p = -y_1∫ $\frac{y_2 f}{W}$ dx + y_2∫ $\frac{y_1 f}{W}$ dx, \] where $f$ is the right side after dividing the equation by the coefficient of $y''$ .; The Wronskian & Variation of Parameters › Variation of Parameters, Worked in Full
The Frobenius method: $x = 0$ is a regular singular point of $x^2y'' + x\,p(x)\,y' + q(x)\,y = 0$ when $p$ and $q$ have power series at $0$ .; Frobenius & Special Functions › Frobenius Series and the Indicial Roots
The guess, family by family: For $y'' + by' + cy = f$ :; Nonhomogeneous Equations & Resonance › Exponential, Sinusoidal and Product Forcing
The guessing table, and the logic behind it: Undetermined coefficients works because differentiation maps each function family into itself: polynomials to polynomials, $e^{kx}$ to itself, $\{\cos\omega x, \sin\omega x\}$ into each other.; Nonhomogeneous Equations & Resonance › Going Deeper: Explanations & Worked Problems
The heat and wave equations, mode by mode: Separation of variables turns a PDE into the eigenvalue problem plus simple time behavior.; Boundary Values & Fourier Series › From Boundary Conditions to Fourier
The impulse: δ and what it does: The Dirac impulse $\delta(t - a)$ models a hammer blow: zero except at one instant, total area $1$ , transform $\mathcal L\{\delta(t-a)\} = e^{-as}$ .; Laplace II: Steps, Impulses & Convolution › Discontinuous and Impulsive Forcing
The ladder of methods: Euler: follow the tangent, $y_{n+1} = y_n + hf(t_n, y_n)$ — global error $O(h)$ .; Numerical Methods: Euler to Runge–Kutta › Computing Solutions Honestly
The logistic solution and its landmarks: The solution of $P' = rP\left(1 - \tfrac PK\right)$ , $P(0) = P_0$ , is \[ P(t) = K1 + Ae^-rt, A = $\frac{K - P_0}{P_0}$ .; Models & Qualitative Behavior › Logistic Growth in Numbers
The pipeline for a second-order IVP: For $ay'' + by' + cy = f(t)$ with $y(0) = y_0$ and $y'(0) = y_1$ , transforming gives \[ (as^2 + bs + c) Y(s) = F(s) + a(sy_0 + y_1) + by_0, \] so $Y = \dfrac{F(s) + a(sy_0 + y_1) + by_0}{as^2 + bs + c}$ : the characteristic polynomial sits in the denominato…; Laplace Transforms › Second-Order Problems and Step Responses
The recurrence method, slowly: Write $y = \sum_{n\ge0} a_nx^n$ and differentiate term by term: $y' = \sum_{n\ge1} na_nx^{n-1}$ , $y'' = \sum_{n\ge2} n(n-1)a_nx^{n-2}$ .; Series Solutions › Going Deeper: Explanations & Worked Problems
The s-shift, and the three kinds of pieces: $\mathcal L\{e^{at}f(t)\} = F(s - a)$ : a transform in which $s$ appears only as $s - a$ inverts to $e^{at}$ times the inverse of the unshifted function.; Laplace Transforms › Partial Fractions and the Shift Rule
The separation recipe: For $u_t = ku_{xx}$ or $u_{tt} = c^2u_{xx}$ on $[0, L]$ with ends held at $0$ :; Boundary Values & Fourier Series › Solving the Heat and Wave Equations
The step-function recipe: To solve a linear constant-coefficient equation whose forcing changes formula at certain times:; Laplace II: Steps, Impulses & Convolution › Piecewise Forcing, Step by Step
The transform and the short table: $\mathcal L\{f\}(s) = \int_0^\infty f(t)e^{-st}dt$ .; Laplace Transforms › The Transform Method
The transform pipeline, and where each rule enters: Solving an IVP by Laplace is a fixed pipeline.; Laplace Transforms › Going Deeper: Explanations & Worked Problems
The Wronskian certifies independence: For solutions $y_1, y_2$ of the same second-order linear homogeneous equation, the Wronskian $W = y_1y_2' - y_1'y_2$ is either never zero or identically zero (Abel's theorem: $W = Ce^{-\int p\,dx}$ ).; The Wronskian & Variation of Parameters › The General Machinery
Three substitutions and how to spot them: Exact Equations & Substitutions › Substitutions in Practice
Try y = xʳ: For $x^2y'' + \alpha xy' + \beta y = 0$ on $x > 0$ , substituting $y = x^r$ gives the indicial equation $r(r-1) + \alpha r + \beta = 0$ .; Frobenius & Special Functions › Cauchy–Euler Equations: Three Cases
Two free coefficients, two solutions: A point $x_0$ is ordinary for $y'' + p(x)y' + q(x)y = 0$ when $p$ and $q$ have convergent power series there.; Series Solutions › Second-Order Equations at Ordinary Points
Two substitutions that unlock whole families: Homogeneous type $y' = f\!\left(\tfrac yx\right)$ : substitute $v = \tfrac yx$ (so $y' = v + xv'$ ) and the equation separates in $v$ and $x$ .; Exact Equations & Substitutions › Exactness and the Great Substitutions
Unit steps carry delays: The Heaviside step $u(t-a)$ switches on at $t = a$ ; any piecewise forcing is a sum of stepped pieces.; Laplace II: Steps, Impulses & Convolution › Discontinuous and Impulsive Forcing
Variation of parameters: the universal particular solution: For $y'' + p y' + q y = f$ with fundamental pair $y_1, y_2$ : seek $y_p = u_1y_1 + u_2y_2$ with functions $u_i$ .; The Wronskian & Variation of Parameters › The General Machinery
When a quantity is conserved: If $E(x, y)$ satisfies $E_x f + E_y g = 0$ for the system $x' = f$ , $y' = g$ , every trajectory stays on a level curve $E = C$ .; Nonlinear Systems & Linearization › Conserved Quantities and Closed Orbits
When eigenvectors run short: If $\lambda$ is a double eigenvalue but $A - \lambda I$ has only one eigenvector line, spanned by $\mathbf v$ , the second solution is \[ u_2 = e^λ t(t v + w), (A - λ I) w = v, \] and $ u = c_…; Systems of Differential Equations › Repeated Eigenvalues
When the test fails: integrating factors: A non-exact equation can often be repaired by multiplying through by $\mu$ .; Exact Equations & Substitutions › Exactness and the Great Substitutions
Where a solution lives: A solution of an initial-value problem is a function on an interval containing the initial point.; First-Order Equations › Separable Equations: Lost Solutions and Blow-Up
Why two constants, and how initial data pins them: A second-order equation reaches back two derivatives, so a solution is not determined until you specify both a starting value $y(0)$ and a starting slope $y'(0)$ — think of a mass on a spring: where it starts and how fast it is moving are independent facts.; Second-Order Linear Equations › Going Deeper: Explanations & Worked Problems
Writing A cos + B sin as one wave: Any $A\cos\omega t + B\sin\omega t$ equals $R\cos(\omega t - \varphi)$ , where $R = \sqrt{A^2 + B^2}$ , $\cos\varphi = \tfrac AR$ and $\sin\varphi = \tfrac BR$ .; Second-Order Linear Equations › Amplitude, Phase and the Decaying Envelope