Differential equation - Revision history

Rongzhou at 11:56, 7 September 2026

2026-09-07T11:56:20Z

RonzzWikiCowriterAI: AI-assisted (RonzzWikiCowriter): present Method 2 as the method of characteristics in general (general quasilinear first-order form, characteristic system, general algebraic solution u = F(ψ(x,t))), with the transport equation and the wave equation derived from it as examples rather than being the method itself. (via update-page on MediaWiki MCP Server)

2026-09-05T20:02:56Z

AI-assisted (RonzzWikiCowriter): present Method 2 as the method of characteristics in general (general quasilinear first-order form, characteristic system, general algebraic solution u = F(ψ(x,t))), with the transport equation and the wave equation derived from it as examples rather than being the method itself. (via update-page on MediaWiki MCP Server)

← Older revision		Revision as of 20:02, 5 September 2026
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	=== Partial differential equations ===		=== Partial differential equations ===

	Two ~~elementary classes of linear PDE admit~~ closed-form solutions: separable problems on bounded domains, ~~by separation~~ of ~~variables~~, ~~and transport and~~ wave ~~problems~~, ~~by travelling waves~~.		Two standard techniques give closed-form solutions of linear PDEs: separation of variables, for separable problems on bounded domains, and the method of characteristics, for first-order equations. The wave equation is also solved by the second technique, because its operator factors into two first-order parts. Both methods below are stated in general and then applied to a concrete equation.

	==== Method 1: separation of variables ====		==== Method 1: separation of variables ====
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	so after one hour $u\approx 100e^{-0.82}\approx 44\,^{\circ}\mathrm{C}$, and $50\,^{\circ}\mathrm{C}$ is reached at $t=\ln 2/(2.3\times10^{-4})\approx 3050\ \text{s}\approx 51$ min.<ref>{{#cite:Q1579}}</ref>		so after one hour $u\approx 100e^{-0.82}\approx 44\,^{\circ}\mathrm{C}$, and $50\,^{\circ}\mathrm{C}$ is reached at $t=\ln 2/(2.3\times10^{-4})\approx 3050\ \text{s}\approx 51$ min.<ref>{{#cite:Q1579}}</ref>

	==== Method 2: ~~travelling waves (~~method of characteristics) ====		==== Method 2: the method of characteristics ====

	'''General form.''' The ~~transport equation for a quantity $u(x,t)$ carried~~ by ~~a flow at constant speed $c$ is~~ the first-order ~~linear PDE~~		'''General form.''' The method of characteristics solves first-order PDEs by tracing curves along which the PDE reduces to ordinary differential equations. In two independent variables the general quasilinear first-order equation is

	$$~~u_t~~+c\,~~u_x~~=0$$		$$A(x,t,u)\,u_x+B(x,t,u)\,u_t=C(x,t,u)$$

	~~'''General algebraic~~ solution~~.''' For any differentiable profile $f$, the travelling wave~~ $u(x,t)=f(x-ct)$ ~~satisfies the equation identically~~, ~~since~~ $~~u_t=-cf'~~(~~x-ct~~)$ ~~and~~ $u_x~~=f'(x-ct)~~$~~; conversely~~, ~~every~~ solution is of ~~this form~~, ~~with~~ $~~f$ fixed by~~ the ~~initial data:~~		A solution $u=u(x,t)$ is a surface in $(x,t,u)$-space. Its tangent plane at each point is spanned by $(1,0,u_x)$ and $(0,1,u_t)$, so a vector $(A,B,C)$ is tangent to the surface exactly when $C=A u_x+B u_t$, the condition expressed by the PDE itself. The solution surface is therefore swept out by the integral curves of the vector field $(A,B,C)$, the characteristic curves, which solve the characteristic system of ordinary differential equations

	$$u(x,t)=f(x~~-ct~~),\qquad u(x,~~0)=f(x~~)$$		$$\frac{dx}{ds}=A(x,t,u),\qquad \frac{dt}{ds}=B(x,t,u),\qquad \frac{du}{ds}=C(x,t,u)$$

	~~so $u$~~ is ~~constant on each~~ characteristic ~~line~~ $x~~-ct=\text{const}$~~, ~~and the initial profile is transported unchanged at speed $c$. For the forced equation $u_t+cu_x~~=s(x,t)$, the ~~source is accumulated along~~ the ~~characteristic:~~		Given data on a curve that is not itself characteristic, such as $u(x,0)=u_0(x)$, one characteristic issues from each point of the curve, and integrating the system carries the data across the region the characteristics cover. For the linear homogeneous case

	$$u(x,t)~~=f(x-ct)~~+~~\int_0^t s\bigl~~(x~~-c(~~t~~-\tau),\tau\bigr~~)\,~~d\tau~~$$		$$a(x,t)\,u_x+b(x,t)\,u_t=0$$

	'''Example: transport of a pollutant.''' A river flows steadily at speed $c=2\ \mathrm{m\,s^{-1}}$, and a factory releases a concentrated slug of pollutant at one point; as long as mixing and diffusion are negligible, the current simply carries the whole slug downstream without changing it. The concentration obeys ~~the transport equation~~ $u_t+2u_x=0$ with the Gaussian initial profile		the $x$- and $t$-equations do not involve $u$, and the third gives $du/ds=0$: the solution is constant along each characteristic. The characteristics form a one-parameter family; let $\psi(x,t)=\text{const}$ be a first integral, a function constant on each member of the family.

			'''General algebraic solution.''' Since $u$ is constant on every characteristic and the characteristics are the level sets of $\psi$, the general solution is an arbitrary function of the first integral,

			$$u(x,t)=F\bigl(\psi(x,t)\bigr)$$

			with $F$ fixed by the initial data. When the right-hand side of the PDE is nonzero, $u$ changes along a characteristic at the rate $C$ (or of the given source term), so the general solution acquires an integral of that term along the curve.

			'''Example: transport of a pollutant.''' For constant coefficients $c$ the equation $u_t+c\,u_x=0$ has characteristics $dx/dt=c$, the straight lines $x-ct=\text{const}$; hence $\psi=x-ct$, and the general algebraic solution is the travelling wave

			$$u(x,t)=F(x-ct),\qquad u(x,0)=F(x)$$

			A river flows steadily at speed $c=2\ \mathrm{m\,s^{-1}}$, and a factory releases a concentrated slug of pollutant at one point; as long as mixing and diffusion are negligible, the current simply carries the whole slug downstream without changing it. The concentration obeys $u_t+2u_x=0$ with the Gaussian initial profile

	$$u(x,0)=50\,e^{-(x/10)^2}\ \mathrm{mg\,L^{-1}}$$		$$u(x,0)=50\,e^{-(x/10)^2}\ \mathrm{mg\,L^{-1}}$$
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	$$u(x,t)=50\,e^{-((x-2t)/10)^2}\ \mathrm{mg\,L^{-1}}$$		$$u(x,t)=50\,e^{-((x-2t)/10)^2}\ \mathrm{mg\,L^{-1}}$$

	After one minute the peak has moved from $x=0$ to $x=ct=120\ \mathrm{m}$, still reading $50\ \mathrm{mg\,L^{-1}}$; pure transport does not spread the slug, which would require the second-order term $\alpha u_{xx}$ of the heat equation.		After one minute the peak has moved from $x=0$ to $x=ct=120\ \mathrm{m}$, still reading $50\ \mathrm{mg\,L^{-1}}$; pure transport does not spread the slug, which would require the second-order term $\alpha u_{xx}$ of the heat equation. With a source $q(x,t)$, the value accumulates along each characteristic:

			$$u(x,t)=F(x-ct)+\int_0^t q\bigl(x-c(t-\tau),\tau\bigr)\,d\tau$$

			'''Example: the wave equation.''' The wave equation

			$$u_{tt}=c^2u_{xx}$$

			is second order, yet its operator factors into two first-order transport operators, so the method of characteristics still applies. Introduce the characteristic coordinates

			$$\xi=x-ct,\qquad \eta=x+ct$$

	~~'''Example:~~ the ~~wave equation.''' The second~~-~~order wave equation is the two~~-~~directional version of~~ the ~~same idea~~, and ~~its~~ general algebraic solution ~~is a pair of waves travelling left and right~~ (d'Alembert, 1747):		in which the operator becomes $u_{tt}-c^2u_{xx}=-4c^2u_{\xi\eta}$, so the equation reads $u_{\xi\eta}=0$. Hence $u_\xi$ depends on $\xi$ alone, and one further integration gives the general algebraic solution (d'Alembert, 1747):

	$$~~u_{tt}=c^2u_{xx}\;\Longrightarrow\;~~u(x,t)=f(x-ct)+g(x+ct)$$		$$u(x,t)=f(x-ct)+g(x+ct)$$

	~~The term $f(x-ct)$ obeys the transport equation with speed $c$ and $g(x+ct)$ with speed $-c$;~~ each ~~therefore satisfies the wave equation, and linearity superposes them into the general solution~~. The functions $f~~$ and $~~g$ are fixed by the initial displacement and velocity: for a string released from rest with initial displacement $\phi(x)$, the conditions $u(x,0)=\phi(x)$ and $u_t(x,0)=0$ give $f=g=\phi/2$, so		a superposition of two travelling waves, one in each direction. The functions $f,g$ are fixed by the initial displacement and velocity: for a string released from rest with initial displacement $\phi(x)$, the conditions $u(x,0)=\phi(x)$ and $u_t(x,0)=0$ give $f=g=\phi/2$, so

	$$u(x,t)=\frac{\phi(x-ct)+\phi(x+ct)}{2}$$		$$u(x,t)=\frac{\phi(x-ct)+\phi(x+ct)}{2}$$

RonzzWikiCowriterAI: AI-assisted (RonzzWikiCowriter): drop the compound-interest numeric scenario (no bank specifies interest via a differential equation), replacing it with a radioactive-decay (carbon-14) scenario; replace the remaining finance example in first-order Method 3 with a harvested fish stock; restructure both PDE methods so that a general algebraic form and general algebraic solution come first, with the heat and wave equations presented afterwards as examples. (via update-page on MediaWiki MCP Server)

2026-09-05T19:58:16Z

AI-assisted (RonzzWikiCowriter): drop the compound-interest numeric scenario (no bank specifies interest via a differential equation), replacing it with a radioactive-decay (carbon-14) scenario; replace the remaining finance example in first-order Method 3 with a harvested fish stock; restructure both PDE methods so that a general algebraic form and general algebraic solution come first, with the heat and wave equations presented afterwards as examples. (via update-page on MediaWiki MCP Server)

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RonzzWikiCowriterAI: AI-assisted (RonzzWikiCowriter): restore scenario introductions to the numerical examples and make each example illustrate the solving process — the scenario is first modelled by a differential equation with concrete numbers, which the method then solves step by step (e.g. the compound-interest scenario now explicitly solves dy/dt = 0.05y, y(0) = 1000). (via update-page on MediaWiki MCP Server)

2026-09-05T19:46:43Z

AI-assisted (RonzzWikiCowriter): restore scenario introductions to the numerical examples and make each example illustrate the solving process — the scenario is first modelled by a differential equation with concrete numbers, which the method then solves step by step (e.g. the compound-interest scenario now explicitly solves dy/dt = 0.05y, y(0) = 1000). (via update-page on MediaWiki MCP Server)

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RonzzWikiCowriterAI: AI-assisted (RonzzWikiCowriter): compress the solution-methods section to derivations in display math with minimal prose; rewrite the history section in a formal academic register, restoring semantic person links. (via update-page on MediaWiki MCP Server)

2026-09-05T19:37:54Z

AI-assisted (RonzzWikiCowriter): compress the solution-methods section to derivations in display math with minimal prose; rewrite the history section in a formal academic register, restoring semantic person links. (via update-page on MediaWiki MCP Server)

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RonzzWikiCowriterAI: AI-assisted (RonzzWikiCowriter): make the homogeneous/particular structure explicit — state the y = y_p + y_h result in the classification section and label homogeneous and non-homogeneous cases inside Methods 2 and 3 (first order) and Method 2 (second order), adding a forced-equation demonstration on the same operator; add PDE Method 2, travelling waves via the method of characteristics (transport equation), with a river-pollutant numeric example and the wave-equation remark. (via update-pag...

2026-09-05T19:29:38Z

AI-assisted (RonzzWikiCowriter): make the homogeneous/particular structure explicit — state the y = y_p + y_h result in the classification section and label homogeneous and non-homogeneous cases inside Methods 2 and 3 (first order) and Method 2 (second order), adding a forced-equation demonstration on the same operator; add PDE Method 2, travelling waves via the method of characteristics (transport equation), with a river-pollutant numeric example and the wave-equation remark. (via update-pag...

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RonzzWikiCowriterAI: AI-assisted (RonzzWikiCowriter): add first-order Method 3 — constant-coefficient linear equations solved by exponential trial and undetermined coefficients (y = C e^{bx} guess, particular-solution guess), with a worked account-with-withdrawals numeric example; note the remaining first-order specialisations; thread the trial through the second-order section and the decision diagram. (via update-page on MediaWiki MCP Server)

2026-09-05T19:21:10Z

AI-assisted (RonzzWikiCowriter): add first-order Method 3 — constant-coefficient linear equations solved by exponential trial and undetermined coefficients (y = C e^{bx} guess, particular-solution guess), with a worked account-with-withdrawals numeric example; note the remaining first-order specialisations; thread the trial through the second-order section and the decision diagram. (via update-page on MediaWiki MCP Server)

@@ Line 67: / Line 67: @@
 == Solving differential equations ==
-There is no formula that solves every differential equation. The practical route is to recognise the class of the equation and apply that class's method. For first-order ODEs there are two standard classes, separable and linear; for second-order ODEs there are two more, equations reducible to direct integration and linear equations with constant coefficients; for linear PDEs on simple domains the standard tool is separation of variables. Each method below is stated for its general case and then applied to a concrete numerical example. When an equation fits none of these classes, it is treated by the numerical, series, or qualitative methods summarised at the end of this section.
+There is no formula that solves every differential equation. The practical route is to recognise the class of the equation and apply that class's method. For first-order ODEs there are two standard classes, separable and linear; when the coefficients of a linear equation are constant, the solution is found even faster by an exponential trial solution (Method 3 below), the same idea that solves the constant-coefficient equations of order two. For second-order ODEs there are two further classes, equations reducible to direct integration and linear equations with constant coefficients; for linear PDEs on simple domains the standard tool is separation of variables. Each method below is stated for its general case and then applied to a concrete numerical example. When an equation fits none of these classes, it is treated by the numerical, series, or qualitative methods summarised at the end of this section.
 === First-order ODEs ===
@@ Line 158: / Line 158: @@
 After 5 seconds $v=49(1-e^{-1})\approx 31\ \mathrm{m\,s^{-1}}$; after 10 seconds $v=49(1-e^{-2})\approx 42\ \mathrm{m\,s^{-1}}$; the terminal $49\ \mathrm{m\,s^{-1}}$ is approached but never quite reached.<ref>{{#cite:Q1576}}</ref>
 === Second-order ODEs ===
@@ Line 185: / Line 213: @@
 $$y''+a\,y'+b\,y=0$$
-models a mass on a spring, a small-angle pendulum, and an RLC circuit. Its solutions are exponentials: try $y=e^{rx}$. Since $y'=re^{rx}$ and $y''=r^2e^{rx}$, substitution gives
+models a mass on a spring, a small-angle pendulum, and an RLC circuit. Its solutions are exponentials (the exponential trial already used in Method 3): try $y=e^{rx}$, since $y'=re^{rx}$ and $y''=r^2e^{rx}$,
 {{#content:Q1644}}
@@ Line 290: / Line 318: @@
 else (no)
    if (First order and linear?\ny' + p(x) y = q(x)?) then (yes)
-     :Integrating factor\nμ = e^{∫ p dx};
+     if (Constant coefficients?\ny' + a y = q(x)?) then (yes)
    else (no)
      if (Second order, of the form\ny'' = f(x)?) then (yes)

RonzzWikiCowriterAI: AI-assisted (RonzzWikiCowriter): reorganise the analytical-methods section by class — first-order ODE methods (separable, integrating factor), second-order ODE methods (direct integration, constant coefficients), PDE methods (separation of variables) — each with general case then concrete numeric example; add a falling-with-air-resistance example under the integrating factor. (via update-page on MediaWiki MCP Server)

2026-09-05T19:13:23Z

AI-assisted (RonzzWikiCowriter): reorganise the analytical-methods section by class — first-order ODE methods (separable, integrating factor), second-order ODE methods (direct integration, constant coefficients), PDE methods (separation of variables) — each with general case then concrete numeric example; add a falling-with-air-resistance example under the integrating factor. (via update-page on MediaWiki MCP Server)

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RonzzWikiCowriterAI: AI-assisted (RonzzWikiCowriter): restore the full history section and strip the narrator prose that restates what the demonstrations show; derivations and worked examples carry the explanation. (via update-page on MediaWiki MCP Server)

2026-09-05T19:01:59Z

AI-assisted (RonzzWikiCowriter): restore the full history section and strip the narrator prose that restates what the demonstrations show; derivations and worked examples carry the explanation. (via update-page on MediaWiki MCP Server)

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RonzzWikiCowriterAI: AI-assisted (RonzzWikiCowriter): condense the article — cut prose commentary, let the derivations and worked examples carry the explanation; keep all semantic embeds, images, diagram and citations. (via update-page on MediaWiki MCP Server)

2026-09-05T18:55:48Z

AI-assisted (RonzzWikiCowriter): condense the article — cut prose commentary, let the derivations and worked examples carry the explanation; keep all semantic embeds, images, diagram and citations. (via update-page on MediaWiki MCP Server)

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