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differential equations

First & second order ODEs, linear systems, resonance, Fourier series, PDEs & Fourier transforms, Laplace transforms.

first order · autonomous & models · linear systems · second order homogeneous · inhomogeneous · oscillations & resonance · fourier series · pdes & fourier transforms · laplace transforms

First order equations

Setup. An ODE of order \(n\) relates an unknown function to its derivatives: \(y^{(n)}=F\big(t,y,y',\ldots,y^{(n-1)}\big)\). The general solution carries \(n\) arbitrary constants; picking a particular solution requires \(n\) initial conditions \(y(t_0),y'(t_0),\ldots,y^{(n-1)}(t_0)\) (an initial value problem). E.g. \(y''=-g\Rightarrow y=-\tfrac12gt^2+c_1t+c_2\) with \(c_1=y'(0)\), \(c_2=y(0)\). Equations for functions of several variables involving partial derivatives are PDEs; single-variable equations are ordinary. Any claimed solution is verified by substituting it and its derivatives into the equation.

Slope fields. \(\dfrac{\dd y}{\dd x}=f(x,y)\) prescribes the slope \(f(x,y)\) of the solution graph at every point; solutions trace integral curves, everywhere tangent to the field. Sketch by isoclines: the curves \(f(x,y)=C\) (level curves of the slope function), along which all segments have slope \(C\). E.g. \(y'=-x/y\): isoclines are lines through the origin, integral curves are circles \(x^2+y^2=R^2\), solutions \(y=\pm\sqrt{R^2-x^2}\) — defined only on \([-R,R]\), ending where the tangent turns vertical.

Existence & uniqueness. If \(f\) is continuous near \((x_0,y_0)\), the IVP \(y'=f(x,y)\), \(y(x_0)=y_0\) has a solution on some interval \((x_0-\varepsilon,\ x_0+\varepsilon)\) — not necessarily extendable further. If \(\partial f/\partial y\) is also continuous, the solution is unique. Failure case: \(y'=2\sqrt y\), \(y(0)=0\) has two solutions \(y=0\) and \(y=x^2\) (\(\partial f/\partial y\) discontinuous at \(y=0\)). Second order analogue: \(y''=F(t,y,y')\) with \(F\) and its partials continuous has exactly one local solution per pair \(y(t_0),y'(t_0)\); likewise for systems.

Separable. \(\dfrac{\dd y}{\dd x}=f(x)g(y)\). First find the equilibrium (constant) solutions \(y=y_0\) with \(g(y_0)=0\); then for \(g(y)\ne0\) divide and integrate: \(\int\dfrac{\dd y}{g(y)}=\int f(x)\dd x+C\), then solve for \(y\). Keep absolute values from log integrals (\(\ln|y-a|\)); after simplifying, the lost equilibrium solutions usually reappear as an allowed constant value (e.g. \(C=0\)). Rational \(1/g\) integrates by partial fractions — coefficients fastest by the Heaviside cover-up method (substitute the root that kills the other terms).

Linear. \(y'+P(x)\,y=Q(x)\) solves by an integrating factor \(I\) chosen so that \(I'=IP\), turning the left side into an exact derivative:

Work the steps rather than quoting the final formula — a wrong \(I\) betrays itself when the reverse product rule fails. Constant-coefficient cases are also separable and give the same answers with less case analysis on \(C\).

Homogeneous substitution. \(\dfrac{\dd y}{\dd x}=F(y/x)\): set \(y=vx\), so \(\dfrac{\dd y}{\dd x}=v+x\dfrac{\dd v}{\dd x}\); the equation for \(v(x)\) is separable.

Euler's method. Numerical stepping for \(y'=f(x,y)\): \(x_{n+1}=x_n+h\), \(y_{n+1}=y_n+h\,f(x_n,y_n)\) — follow the slope field in straight steps of size \(h\).

Series solutions. Two routes. (1) Repeated differentiation: differentiate the equation implicitly to get \(y'',y''',\ldots\) in terms of lower derivatives, evaluate at \(x_0\) starting from the initial data, and assemble Taylor's series \(y(x)=\sum y^{(n)}(x_0)\,(x-x_0)^n/n!\). (2) Substitution: insert \(y=\sum a_nx^n\) (or \(\sum c_nx^n/n!\) with \(c_n=y^{(n)}(0)\), often cleaner), match coefficients of each power, and solve the resulting recursion — writing the first few terms by hand to see where the recursion starts. E.g. \(y'=xy\Rightarrow(n+1)a_{n+1}=a_{n-1}\Rightarrow y=a_0e^{x^2/2}\). Constant-coefficient recursions \(c_{n+2}=pc_{n+1}+qc_n\) solve as \(c_n=A\lambda_1^n+B\lambda_2^n\) with \(\lambda_i\) the roots of \(\lambda^2=p\lambda+q\) (the eigenvalues of the companion matrix); \(A,B\) from \(c_0,c_1\). Both routes work for second order equations too — every \(y^{(k)}(t_0)\) is determined by \(y(t_0),y'(t_0)\), which is why those two numbers pin down the solution.

Autonomous equations & models

Autonomous equations. \(\dfrac{\dd y}{\dd t}=f(y)\) (right side independent of \(t\)). Equilibrium solutions: \(y(t)=y_0\) constant wherever \(f(y_0)=0\). Between equilibria the sign of \(f\) decides everything: where \(f(y)\gt0\) solutions increase, where \(f(y)\lt0\) they decrease — monotonically toward an equilibrium value or off to infinity. A sign plot of \(f\) therefore gives the full qualitative picture. An equilibrium is stable if nearby solutions are drawn back and approach it as \(t\to\infty\), unstable otherwise. Time-translation invariance: if \(y(t)\) solves, so does \(y(t-t_0)\) — solutions starting later just replay shifted.

Growth & decay. \(y'=ky\Rightarrow y=Ce^{kt}\) (exponential growth \(k\gt0\), decay \(k\lt0\)). Approach to equilibrium \(A\): \(y'=k(y-A)\Rightarrow y=A+Ce^{kt}\), stable iff \(k\lt0\). Newton's law of cooling is the stable case: \(T'=-k\,(T-T_{\mathrm{env}})\Rightarrow T=T_{\mathrm{env}}+Ce^{-kt}\to T_{\mathrm{env}}\).

Logistic. \(n'=kn(a-n)\Rightarrow n(t)=\dfrac{a}{1+Ce^{-akt}}\) (separate, integrate by partial fractions). Equilibria \(n=0\) (unstable) and \(n=a\) (stable, the carrying value); solutions between rise sigmoidally toward \(a\).

First order linear systems

Systems & velocity fields. \(\dfrac{\dd x}{\dd t}=P(x,y)\), \(\dfrac{\dd y}{\dd t}=Q(x,y)\): the solution \((x(t),y(t))\) moves like a fluid particle in the velocity field \(P\,\hat\imath+Q\,\hat\jmath\) — the field gives direction and speed, not just slope. Equilibrium points are where the field vanishes (\(P=Q=0\)); they are constant solutions. Trajectories lie on integral curves solving \(\dfrac{\dd y}{\dd x}=\dfrac{Q(x,y)}{P(x,y)}\). Existence & uniqueness holds as for scalar equations. Only 2D autonomous systems are treated: higher dimensions and time-dependent 2D systems can be chaotic.

Linear systems. \(\dfrac{\dd\vv x}{\dd t}=A\vv x+\vv b\); homogeneous iff \(\vv b=\vv 0\). Superposition: linear combinations \(c_1\vv x_1+c_2\vv x_2\) of homogeneous solutions solve the homogeneous system; inhomogeneous general solution \(=\) one particular solution \(+\) general homogeneous solution. For invertible \(A\) a particular solution is the equilibrium \(\vv x_{\mathrm{eq}}=-A^{-1}\vv b\), so \(\vv x(t)=\vv x_{\mathrm{eq}}+c_1\vv x_1(t)+c_2\vv x_2(t)\), with \(c_1,c_2\) fixed by \(\vv x(0)\).

Eigenvalue method. Along an eigenvector the field is parallel to the position, so try \(\vv x=f(t)\vv v\) with \(A\vv v=\lambda\vv v\); then \(f'=\lambda f\), giving straight-line solutions \(e^{\lambda t}\vv v\). With a real eigenbasis:

Relative exponential rates set the geometry: as \(t\to\infty\) the term with the larger \(\lambda\) dominates and trajectories run parallel to its eigenline; as \(t\to-\infty\) the smaller-\(\lambda\) term dominates, so near the origin trajectories run tangent to the slow eigendirection.

Complex eigenvalues. \(\lambda=p\pm iq\) (\(q\gt0\)) with complex eigenvector \(\vv v=\vv v_1+i\vv v_2\): the complex solution \(e^{\lambda t}\vv v\) splits into two real solutions \(\vv x_1=e^{pt}(\vv v_1\cos qt-\vv v_2\sin qt)\), \(\vv x_2=e^{pt}(\vv v_1\sin qt+\vv v_2\cos qt)\) (real and imaginary parts each solve, by linearity). Trajectories wind around the origin with period \(2\pi/q\): spiral inward if \(p\lt0\), outward if \(p\gt0\), closed ellipses (aligned with \(\vv v_1,\vv v_2\) — a linear image of circles) if \(p=0\). Orientation: compute one velocity vector and see which way it points.

Repeated eigenvalue. One straight-line solution \(e^{\lambda t}\vv v\); a second of the form \(e^{\lambda t}(\vv w+t\vv v)\) with \((A-\lambda I)\vv w=\vv v\).

Stability. \(\vv x_{\mathrm{eq}}\) is stable — every solution decays to it — iff every eigenvalue of \(A\) has negative real part (real case: both \(\lambda\lt0\); complex case: \(p\lt0\)). True for linear autonomous systems of any size.

Second order homogeneous equations

Linear operators. A linear ODE is \(p_n(t)y^{(n)}+\cdots+p_1(t)y'+p_0(t)y=f(t)\), i.e. \(Oy=f\) for a linear differential operator \(O=p_nD^n+\cdots+p_1D+p_0\), \(D=\dfrac{\dd}{\dd t}\). Linearity \(O[c_1y_1+c_2y_2]=c_1Oy_1+c_2Oy_2\) yields the three facts everything rests on: (1) homogeneous solutions superpose; (2) \(y_p+y_h\) solves \(Oy=f\) whenever \(Oy_p=f\), \(Oy_h=0\) — and every solution has this form; (3) if \(Oy_1=f_1\), \(Oy_2=f_2\) then \(O[c_1y_1+c_2y_2]=c_1f_1+c_2f_2\), so responses to a sum of inputs superpose.

Characteristic equation. For constant coefficients \(y''+py'+qy=0\), the exponential ansatz \(y=e^{\lambda t}\) (using \(D^ke^{\lambda t}=\lambda^ke^{\lambda t}\)) reduces the ODE to the auxiliary (characteristic) equation \(\lambda^2+p\lambda+q=0\). Three cases: distinct real roots \(\Rightarrow y=c_1e^{\lambda_1t}+c_2e^{\lambda_2t}\); complex roots \(a\pm ib\Rightarrow y=e^{at}(c_1\cos bt+c_2\sin bt)\) (real and imaginary parts of \(e^{(a+ib)t}\), each a real solution by linearity); repeated root \(\lambda\Rightarrow y=(c_1+c_2t)e^{\lambda t}\). Initial conditions \(y(t_0),y'(t_0)\) fix \(c_1,c_2\) via two linear equations.

Operator factoring. Constant-coefficient operators commute and factor like polynomials: \(D^2+pD+q=(D-\lambda_1)(D-\lambda_2)\) (false for variable coefficients — \((D+t)^2\ne D^2+2tD+t^2\)). Solving \((D-\lambda_1)(D-\lambda_2)y=0\) by two chained first-order solves (set \(u=(D-\lambda_2)y\), solve \((D-\lambda_1)u=0\), then an integrating factor) is repeated integration: it re-derives the exponential solutions, proves the ansatz captures the general solution, and in the repeated-root case produces the extra \(te^{\lambda t}\).

Inhomogeneous equations

Structure. \(my''+ly'+ky=f(t)\): the whole difficulty is one particular solution \(y_p\); then \(y=y_p+y_h\) with the two homogeneous constants absorbing any initial conditions. Split a composite right side into pieces, solve each, recombine by linearity.

Exponential response. Right side \(e^{rt}\): try \(y_p=Ce^{rt}\); substituting gives \((mr^2+lr+k)C=1\), so \(y_p=\dfrac{e^{rt}}{mr^2+lr+k}\) — valid whenever \(r\) is not a characteristic root (else the denominator vanishes and the ansatz collapses to \(0=e^{rt}\)).

Undetermined coefficients. Applying \(mD^2+lD+k\) to \(p(t)e^{rt}\) (polynomial \(\times\) exponential) returns the same shape, so for right side \(q(t)e^{rt}\) with \(\deg q=n\), the ansatz \(y_p=t^s\,p(t)\,e^{rt}\), \(\deg p=n\), always works, where \(s\in\{0,1,2\}\) is the multiplicity of \(r\) as a characteristic root. Pure exponential case: \(Ae^{rt}\), \(Ate^{rt}\), or \(At^2e^{rt}\). Terms of the ansatz that solve the homogeneous equation contribute nothing and may be dropped (e.g. \(y''-2y'+y=e^t\): \(r=1\) is a double root, ansatz \(At^2e^t\) gives \(A=\tfrac12\)). Resonant cases can also be ground out by operator factoring and repeated integration.

Sinusoidal response — complexify. \(\cos\omega t=\operatorname{Re}e^{i\omega t}\) (Euler), so to solve \(my''+ly'+ky=\cos\omega t\), solve the complexified equation \(mz''+lz'+kz=e^{i\omega t}\) with \(z=Ze^{i\omega t}\) and take \(y_p=\operatorname{Re}z\) (real and imaginary parts of a complex solution solve the equations with right sides \(\cos\omega t\), \(\sin\omega t\) respectively). Equivalent to superposing solutions for \(\tfrac12e^{i\omega t}+\tfrac12e^{-i\omega t}\), since those come in conjugate pairs.

Oscillations & resonance

Harmonic oscillator. Mass on a spring (Newton \(+\) Hooke): \(my''+ky=0\), characteristic roots \(\pm i\omega_0\) with natural frequency \(\omega_0=\sqrt{k/m}\); \(y=a\cos\omega_0t+b\sin\omega_0t\). Phase–amplitude form: \(y=A\cos(\omega_0t-\phi)\) with \(a=A\cos\phi\), \(b=A\sin\phi\), \(A=\sqrt{a^2+b^2}\) (angle-subtraction identity). The inhomogeneous version \(my''+ly'+ky=f(t)\) is a driven oscillator: input \(f\), response \(y\).

Damping. \(my''+ly'+ky=0\) (\(m,l,k\gt0\)): roots \(\lambda=-d\pm\sqrt{d^2-\omega_0^2}\), \(d=\dfrac{l}{2m}\). Overdamped \(d\gt\omega_0\) (\(l^2\gt4mk\)): two negative real roots, non-oscillatory decay. Underdamped \(d\lt\omega_0\): \(y=e^{-dt}(c_1\cos\omega t+c_2\sin\omega t)\), \(\omega=\sqrt{\omega_0^2-d^2}\) — a sinusoid inside the shrinking envelope \(\pm e^{-dt}\); damping lowers the frequency. Critically damped \(d=\omega_0\) (\(l^2=4mk\)): \(y=(c_1+c_2t)e^{-dt}\). In every case the homogeneous solution decays to \(0\).

Forced oscillations. \(my''+ly'+ky=A_d\cos(\omega t)\): complexifying with \(z=Ze^{i\omega t}\) gives the frequency response function \(Z(\omega)=\dfrac{1}{(k-m\omega^2)+il\omega}=G\,e^{-i\phi}\) — amplitude gain \(G(\omega)=\dfrac{1}{\sqrt{(k-m\omega^2)^2+l^2\omega^2}}\), phase shift \(\tan\phi=\dfrac{l\omega}{k-m\omega^2}\). General solution \(y=GA_d\cos(\omega t-\phi)+y_h\): the steady-state response, a sinusoid at the driving frequency with amplitude ratio \(G\) and shifted peaks, plus a transient \(y_h\) that dies off exponentially.

Resonance. \(G\to0\) as \(\omega\to\infty\); \(G\) is maximized at the resonant frequency \(\omega_{\mathrm{res}}=\sqrt{\dfrac{2km-l^2}{2m^2}}=\sqrt{\omega_0^2-2d^2}\), which exists only when \(l^2\lt2km\) — only lightly underdamped oscillators resonate (critical damping is \(l^2=4mk\)); otherwise the gain peaks at \(\omega=0\). Undamped: \(G=\dfrac{1}{k-m\omega^2}\) diverges as \(\omega\to\omega_0\), and at exact resonance the exponential ansatz fails; undetermined coefficients on \(y''+\omega_0^2y=\cos(\omega_0t)\) gives \(y_p=\dfrac{t\sin(\omega_0t)}{2\omega_0}\) — oscillation with linearly growing amplitude. So a bounded input can produce an unbounded (or violently large) response when driven near \(\omega_0\) with too little damping — the engineering catastrophe mode (collapsing bridges, overloaded circuits).

Fourier series

Series & coefficients. A \(2\pi\)-periodic function (\(f(t+2\pi)=f(t)\); any period rescales to \(2\pi\)) decomposes into sinusoids at integer frequencies:

The coefficient formula follows by multiplying by \(e^{-int}\) and integrating: \(\int_{-\pi}^{\pi}e^{i(k-n)t}\dd t=0\) for \(k\ne n\) and \(2\pi\) for \(k=n\), so only one term survives — which also shows the coefficients are unique. \(c_0\) is the mean value of \(f\). Real–complex dictionary (real \(f\)): \(a_n=2\operatorname{Re}c_n\), \(b_n=-2\operatorname{Im}c_n\), \(c_{\pm n}=\tfrac12(a_n\mp ib_n)\); equivalently \(a_n=\dfrac1\pi\int_{-\pi}^{\pi}f\cos nt\dd t\), \(b_n=\dfrac1\pi\int_{-\pi}^{\pi}f\sin nt\dd t\). Simple cases are algebraic: \(\cos^3t=\big(\tfrac{e^{it}+e^{-it}}{2}\big)^3=\tfrac34\cos t+\tfrac14\cos 3t\).

Even & odd. Any function splits uniquely into even \(+\) odd parts (\(e^t=\cosh t+\sinh t\)). Even periodic \(f\): cosine series only, \(a_n=\dfrac2\pi\int_0^{\pi}f\cos nt\dd t\); odd: sine series only, \(b_n=\dfrac2\pi\int_0^{\pi}f\sin nt\dd t\) (integrand is even, so integrate half the range and double). Half-range expansions: a function \(h\) on \([0,\pi]\) extends oddly (sine series — endpoint values exact if \(h(0)=h(\pi)=0\)) or evenly (cosine series — clean if \(h'(0)=h'(\pi)=0\)), then \(2\pi\)-periodically.

Standard examples. Square wave (\(0\) on \((-\pi,0)\), \(\pi\) on \((0,\pi)\)): \(\mathrm{sq}(t)=\dfrac\pi2+2\Big(\sin t+\dfrac{\sin 3t}{3}+\dfrac{\sin 5t}{5}+\cdots\Big)\). Sawtooth \(f(t)=-t\) on \((-\pi,\pi)\): \(b_n=\dfrac{2(-1)^n}{n}\), so \(f=2\sum_{n\ge1}\dfrac{(-1)^n}{n}\sin nt\).

Convergence. If \(f\) is \(2\pi\)-periodic and differentiable except at finitely many points where one-sided limits exist, the series converges to \(f(t_0)\) at every continuity point and to the jump average \(\tfrac12\big(f(t_0^+)+f(t_0^-)\big)\) at discontinuities (the sawtooth series gives \(0\) at \(t=\pi\)). For non-periodic \(f\) the expansion is only valid on \((-\pi,\pi)\).

Parseval. \(\displaystyle\int_{-\pi}^{\pi}|f(t)|^2\dd t=2\pi\sum_{n=-\infty}^{\infty}|c_n|^2\). Applied to \(f(t)=t\) (\(c_n=i(-1)^n/n\), \(c_0=0\)): \(\sum_{n\ge1}\dfrac{1}{n^2}=\dfrac{\pi^2}{6}\).

Frequency domain. Writing \(\hat f_n\) for the coefficients: scaling and addition act coefficient-wise; differentiation is multiplication, \(\widehat{(f')}_n=in\hat f_n\) (no product/chain rules needed); multiplication of functions becomes a convolution of coefficient sequences \(\widehat{(fg)}_n=\sum_k\hat f_k\hat g_{n-k}\). So a periodic-forced ODE becomes algebra: \(my''+ly'+ky=f\Rightarrow\hat y_n=\dfrac{\hat f_n}{k-mn^2+iln}\), and \(y=\sum\hat y_ne^{int}\) converges for \(m,l\gt0\).

Boundary value problems. \(my''+ky=f(t)\) on \([0,\pi]\): with Dirichlet conditions \(y(0)=y(\pi)=0\), expand \(f=\sum b_n\sin nt\) and solve termwise — \(y=\sum\dfrac{b_n}{k-mn^2}\sin nt\) satisfies the boundary conditions automatically. With Neumann conditions \(y'(0)=y'(\pi)=0\), use the cosine series the same way.

PDEs & Fourier transforms

The big three. With subscripts for partials (\(u_t=\partial u/\partial t\)): heat/diffusion equation \(u_t-\alpha^2u_{xx}=\rho(x,t)\) (\(\alpha^2\) the diffusion rate, \(\rho\) a source; models temperature in a rod); wave equation \(u_{tt}-c^2u_{xx}=\rho(x,t)\) (\(c\) the propagation speed); Poisson equation \(\nabla^2\varphi=-\rho(x,y,z)\), \(\nabla^2=\partial_x^2+\partial_y^2+\partial_z^2\) the Laplace operator (e.g. electrostatic potential from charge density; source-free case is Laplace's equation \(\nabla^2\varphi=0\)). 3D heat/wave equations replace \(u_{xx}\) by \(\nabla^2u\); their steady states (\(u_t=0\)) satisfy the Laplace/Poisson equation. Solutions are pinned down by boundary conditions — Dirichlet: prescribe values of \(u\) on the boundary; Neumann: prescribe derivatives (insulated rod ends: \(u_x(0,t)=u_x(L,t)=0\), which conserves total heat \(\int_0^Lu\dd x\)) — plus initial conditions \(u(x,0)=f(x)\) for time-dependent problems.

Separation of variables. For \(u_t=\alpha^2u_{xx}\) try \(u=X(x)T(t)\): then \(\dfrac{T'}{\alpha^2T}=\dfrac{X''}{X}\), and a function of \(t\) alone equalling a function of \(x\) alone forces both to be a constant \(\lambda=-\mu^2\): \(X''=\lambda X\), \(T'=\alpha^2\lambda T\). Dirichlet conditions \(u(0,t)=u(L,t)=0\) kill all but \(\mu=\dfrac{n\pi}{L}\), giving modes \(u_n=e^{-\alpha^2(n\pi/L)^2t}\sin\dfrac{n\pi x}{L}\). The equation and boundary conditions are linear homogeneous, so modes superpose; matching \(u(x,0)=f(x)\) is exactly a half-range sine expansion (Fourier's original argument — his claim that every function admits one was vindicated later):

High modes decay fastest — the profile smooths toward equilibrium.

Fourier transform. Non-periodic functions superpose sinusoids of all real frequencies (a periodic function's series in the limit period \(\to\infty\), sums becoming integrals):

Inversion is valid when \(\int|f|^2\dd t\) is finite (\(f\to0\) fast enough at \(\pm\infty\)). \(\mathcal F\) is linear, and differentiation becomes multiplication: \(\mathcal F[f']=ik\hat f\) (integrate by parts). Worked pair: \(f=e^{-|t|}\Rightarrow\hat f=\dfrac{2}{1+k^2}\). ODEs transform to algebra — \(y''+y'+y=e^{-|t|}\Rightarrow\hat y=\dfrac{2}{(1+k^2)\big((1-k^2)+ik\big)}\) — but the method only finds square-integrable solutions (no free initial conditions) and inversion integrals are painful; the Laplace transform is the IVP-adapted fix.

Convolution & the heat kernel. A product of transforms inverts to a convolution: \(\mathcal F^{-1}[\hat f\hat g]=(f*g)(x)=\int_{-\infty}^{\infty}f(v)\,g(x-v)\dd v\). Heat equation on the whole line, \(u(x,0)=f(x)\): transforming in \(x\) gives \(\hat u_t=-\alpha^2k^2\hat u\), an ODE in \(t\), so \(\hat u=\hat f(k)e^{-\alpha^2k^2t}\); inverting, \(u(x,t)=(f*\Phi)(x)\) with the fundamental solution \(\Phi(x,t)=\dfrac{1}{\sqrt{4\pi\alpha^2t}}\,e^{-x^2/(4\alpha^2t)}\) — a unit of heat at the origin spreading as a widening Gaussian.

Laplace transforms

Definition. For \(y(t)\) on \([0,\infty)\):

The integral converges for \(\operatorname{Re}s\) large enough, provided \(y\,e^{-st}\to0\) for some \(s\) (fails for \(e^{t^2}\)). \(\mathcal L\) is linear; for \(f\) vanishing on \(t\lt0\) it generalizes the Fourier transform (\(Y(ik)=\hat y(k)\)). A function on \([0,\infty)\) is uniquely determined by its transform, so \(\mathcal L^{-1}\) makes sense — computed in practice by recognizing entries, not by an inversion integral.

Rules. The exponential shift rule \(\mathcal L\{e^{-at}f\}=F(s+a)\) generates the damped-sinusoid entries from the plain ones; the delay rule \(\mathcal L\{f(t-a)\theta(t-a)\}=e^{-as}F(s)\) handles switched-on inputs; the derivative rule \(\mathcal L\{f'\}=sF-f(0)\) (integration by parts) is what makes initial value problems algebraic — initial data enters the transform itself. \(\mathcal L\{t^n\}=n!/s^{n+1}\) follows by transforming \((t^n/n!)^{(n)}=1\); with the shift rule this covers every polynomial-times-exponential.

Solving IVPs. \(my''+ly'+ky=f(t)\), \(y(0)=y_0\), \(y'(0)=v_0\): transform both sides — \((ms^2+ls+k)Y(s)-(ms+l)y_0-mv_0=F(s)\) — solve for \(Y(s)\) by pure algebra, and invert. E.g. \(y''+2y'+y=e^{-t}\), rest initial conditions: \((s+1)^2Y=\dfrac{1}{s+1}\Rightarrow Y=\dfrac{1}{(s+1)^3}\Rightarrow y=\dfrac{t^2}{2}e^{-t}\).

Inverting rational functions. Factor the denominator and expand in partial fractions. Distinct real roots: \(F=\sum\dfrac{A_i}{s-\lambda_i}\Rightarrow f=\sum A_ie^{\lambda_it}\); each \(A_i\) by cover-up (multiply by \(s-\lambda_i\), set \(s=\lambda_i\)). Complex conjugate pair: complete the square, \((s+a)^2+b^2\), write the term as \(\dfrac{C(s+a)+E}{(s+a)^2+b^2}\), which inverts to \(e^{-at}\big(C\cos bt+\tfrac Eb\sin bt\big)\); coefficients by evaluating at a complex root and matching real/imaginary parts. Root \(\lambda\) of multiplicity \(m\): include the whole ladder \(\dfrac{A_1}{s-\lambda}+\cdots+\dfrac{A_m}{(s-\lambda)^m}\), inverting via \(\mathcal L^{-1}\big\{(s-\lambda)^{-k}\big\}=\dfrac{t^{k-1}}{(k-1)!}e^{\lambda t}\).

Step & impulse. Unit step \(\theta(t-a)\): \(0\) for \(t\lt a\), \(1\) for \(t\gt a\) — an idealized switch. Piecewise functions are combinations of delayed pieces \(g(t-a)\theta(t-a)\) (each jump adds a step term), transformed by the delay rule. The unit impulse \(\delta(t-a)\) is the idealized narrow unit-area spike (\(\theta'\)): a generalized function with \(\delta(t)=0\) for \(t\ne0\), \(\int_a^b\delta(t)\dd t=1\) when \(a\le0\le b\), used legitimately inside integrals via the sifting property \(\int f(t)\,\delta(t-a)\dd t=f(a)\); hence \(\mathcal L\{\delta(t-a)\}=e^{-as}\).

Transfer function. For the oscillator \(my''+ly'+ky=f(t)\) at rest (\(y(0)=y'(0)=0\)), the transform of the response is \(Y(s)=F(s)H(s)\) with transfer function \(H(s)=\dfrac{1}{ms^2+ls+k}\) — in the \(s\)-domain the black box just multiplies. The impulse response \(h=\mathcal L^{-1}\{H\}\) is the output for input \(\delta(t)\) (a unit kick at \(t=a\) gives \(h(t-a)\theta(t-a)\)), and the general response is the convolution

For a sinusoidal input \(e^{i\omega t}\), partial fractions leave transients plus the steady-state term \(H(i\omega)e^{i\omega t}\): the steady-state amplitude is \(|H(i\omega)|=\dfrac{1}{\sqrt{(k-m\omega^2)^2+l^2\omega^2}}\) — the frequency response function again, now read off the transfer function.