Non-homogeneous nth-order differential equations {adjoint equation} can have non-constant coefficients.
dy/dx = p(x) * y + q(x) * y^a, where a is not zero and a is not one {Bernoulli equation}.
y = x * y' + f(y') {Clairaut's equation} {Clairaut equation}.
Y(n,z) = sum from n = 0 to n = infinity of ((-1)^r * (z/2)^(n + 2*r) / (r! * (n + r)!)) * (2 * log(z/2) + 2*c - (sum from m = 1 to m = n + r of (1/m)) - (sum from m = 1 to m = r of (1/m))) - (sum from r = 0 to r = n - 1 of (z/2)^(2*r - n) * (n - r - 1)! / r!) {factorial equation}.
Special functions {Hermite function} can solve ordinary differential equations over infinite or semi-infinite intervals.
(p^2 - h^2) * (r^2 - k^2) * ((d^2)E(r) / (dr)^2) + r * (2 * p^2 - h^2 - k^2) * (dE(r) / dr) + ((h^2 + k^2) * p - n * (n + 1) * r^2) * E(r) = 0, where (d^2) is second derivative, r is radius, p is vertical dimension, n is parameter, and (k,h) is point {Lamé's differential equation} {Lamé differential equation}. Solution functions {Lamé function} are elliptical harmonics of first or second kind.
Linear equation has variables raised only to first power. Differential equation has derivatives {linear differential equation, calculus}. Second-order differential equation has second derivatives. Homogeneous equation has function equal to zero.
homogeneous
Second-order first-degree linear homogeneous differential general equation is a * (d^2)x + b * dx + c = 0, where (d^2) is second differential, d is first differential, x is independent variable, and a, b, and c are coefficients. General solution is c1 * e^(r1 * x) + c2 * e^(r2 * x), where e is base of natural logarithms, c1 and c2 are constants, r1 and r2 are roots, and x is independent variable.
non-homogeneous
Second-order, first-degree linear non-homogeneous differential general equation is a * (d^2)x + b * dx = -c, with same general solution.
Functions can have multiple values. Partial-differential-equation systems can model multiple-value functions {theory of multiple-valued functions} {multiple-valued function theory}.
To solve non-linear differential equations {non-linear differential equation}, look for stable point using qualitative theory or find characteristic equation, using theorems {Poincaré-Bendixson theorem} and operations {Painleve transcendent operations}.
dy/dx = a0(x) + a1(x) * y + a2(x) * y^2 {Riccati equation}.
Second-order ordinary differential equation can expand into infinite series of eigenfunctions {Sturm-Liouville theory, differential equation}.
No exact model exists for three mutually gravitationally interacting bodies {three-body problem}. Approximate solutions model mass-center straight-line motions and use energy and momentum conservation laws.
two dimensions
Perhaps, physical problems in three dimensions can reduce to problems in two dimensions using information concepts. Information is on surface, instead of in volume. Projection onto surface from volume has same information about positions, momenta, and transition probabilities. For special cases, reducing to two dimensions can solve the three-body problem.
Functions {Mathieu function} {Weber function}, in mutually orthogonal curvilinear coordinates, can solve the potential equation.
Circular, elliptic, hyperbolic, and other analytic functions {automorphic function} can generalize to find higher properties.
invariance
Automorphic functions are invariant if z' = (a*z + b) / (c*z + d) where a*d - b*c = 1, z is complex number, and z' is complex conjugate.
theta
theta(z) = sum from i = 0 to i = infinity of (c(i) * z + d(i))^(-2 * m) * H(z(i)), where m > 1 and H is rational function. Automorphic-function groups can be discrete or discontinuous groups of infinite order {theory of automorphic functions, discontinuous} {automorphic function theory, discontinuous}.
(d^2)u / (dr)^2 + du / (r * dr) + a^2 - b^2 / r^2 = 0 {Bessel equation}. x^2 * y'' + x * y' + (x^2 - n^2) * y = 0, where (d^2) is second derivative and x and n are complex, has two solutions. J(n,x) = (1 / (2 * pi)) * (integral from u = 0 to u = 2*pi of (cos(n*u) - x * sin(u)) * du). x * J(n+1,x) - 2 * n * J(n,x) + x * J(n-1,x) = 0.
Partial differential equation {partial differential equation} can have order greater than one, with second or higher derivatives. Partial differential equations of order greater than one are equivalent to first-order partial-differential-equation systems {system of partial differential equations}. For example, homogeneous, linear, second-order partial differential equation can be two first-order partial differential equations. c1 * (D^2)x + c2 * Dx + c3 = 0, where (D^2) is second derivative, D is first derivative, and c1, c2, and c3 are constants. c11 * Dx + c12 = 0 and d21 * Dx + d22 = 0, where D is first derivative and c11, c12, c21, and c22 are constants.
conditions
Partial differential equations can use boundary values and initial values.
Methods {arithmetic means method} {method of arithmetic means} {sweeping out method} {method of sweeping out} similar to ordinary-differential-equation methods can find partial-differential equation-system solutions.
Partial differential equation {heat-flow equation} {heat equation} can represent heat flow. Second derivatives of heat with respect to distance equal constant squared times first partial derivative of heat with respect to time: (D^2)T / Dx + (D^2)T / Dy + (D^2)T / Dz = (k^2) * (DT / Dt), where T is heat, (D^2) is second partial derivative, D is partial derivative, k is constant, and x, y, z, and t are coordinates.
Variable separation on partial differential equations can result in ordinary differential equations that use parameters {eigenfunction}| that have value sequences {eigenvalue, mathematics}. Ordinary differential equation solutions use eigenvalues. Second-order ordinary differential equations can expand into infinite series of eigenfunctions {Sturm-Liouville theory, eigenfunction}.
For homogeneous functions u with n variables, n*u = x * (Du/Dx) + y * (Du/Dy) + ..., where D are partial differentials {Euler's theorem} {Euler theorem}.
First-order partial differential equations {Navier-Stokes equation} describe fluid dynamics, using velocity, pressure, density, and viscosity. Examples are fluid motions and viscous-media object motions.
Partial differential equations {Plateau's problem} {Plateau problem} can represent surfaces of least area under closed boundaries. Example is soap film in loop.
Partial differential equations {total differential equation} can be P*dx + Q*dy + R*dz = 0.
Partial differential equations {excess function} {E-function} can represent energy function.
Energy or force equations can minimize quantities {least constraint principle} {principle of least constraint}. For example, sum of kinetic-energy-to-potential-energy changes over time {action} can be minimum: integral of (kinetic energy - potential energy) * dt.
Partial differential equations {Hamilton-Jacobi equation} can represent potential energy plus kinetic energy equals total energy. Sum of second partial derivatives of potential with respect to each coordinate and partial derivative of potential with respect to time equals zero: (D^2)V / Dx + (D^2)V / Dy + (D^2)V / Dz - DV / Dt = 0, where V is potential, (D^2) is second partial derivative, D is partial derivative, and x, y, z, and t are coordinates.
Operators {Laplace operator} {Laplace's operator}, on vector fields or potentials {del squared of f}, can be second derivatives, describe field-variation smoothness, be vectors, and be non-linear.
potential
Partial differential equations {potential equation} {Laplace's equation} can represent potentials. Potential V depends on distance r from mass or charge center: r = (x^2 + y^2 + z^2)^0.5.
Second partial derivative of potential V with respect to distance along x-axis plus second partial derivative of potential V with respect to distance along y-axis plus second partial derivative of potential V with respect to distance along z-axis equals zero: (D^2)V / Dx + (D^2)V / Dy + (D^2)V / Dz = 0, where (D^2) is second partial derivative, D is partial derivative, and V is constant times distance from center, because dx^2 / dx = 2 * x and d(2*x) / dx = 0.
solution
Spherical functions or Legendre polynomials can solve potential equation.
(1 - x^2) * y'' - 2 * x * y' + n * (n + 1) * y = 0, where n is parameter {Legendre differential equation}. Solutions are polynomials {Legendre polynomial}, potential equation spherical coordinates derived by variable separation, or spherical harmonics of second kind.
For boundaries with potential change zero, calculations can find potential change normal to region {Neumann problem} {second fundamental problem}.
If potential-equation right side equals -4 * pi * (energy density), rather than zero, equation describes gravitation and electrostatics {Poisson's equation} {Poisson equation}. Energy density is pressure.
Functions {periodic function} can solve partial differential equations (D^2)y / Dt = (a^2) * ((D^2)y / Dx), where (D^2) is second partial derivative, D is partial derivative, a is constant, t is time, x is distance, and y is function of time and distance. Representing functions by infinite trigonometric series can solve periodic equations. Parameters can analyze function, so y(t,x) = h(t) * g(x). Parameters set equation eigenfunction and eigenvalues.
First-order partial differential equation {electromagnetic wave equation} describes electromagnetic-wave energy oscillations.
Waves {cylindrical wave} can have partial differential equations. Second partial derivative of velocity with respect to time, times 1/c^2, equals three times partial derivative of velocity with respect to distance along pipe length, times 1/z, plus second partial derivative of velocity with respect to distance: ((D^2)v / Dt) * (1 / c^2) = 3 * (Dv / Dz) * (1/z) + (D^2)v / Dz, where (D^2) is second partial derivative, D is partial derivative, v is velocity, z is distance, t is time, and c is constant.
Waves {spherical wave} can have partial differential equations. Second partial derivative of radial velocity with respect to time, times 1/c^2, equals four times partial derivative of radial velocity with respect to radius, times 1/V, plus second partial derivative of radial velocity with respect to radius: ((D^2)s / Dt) * (1/c^2) = 4 * (Ds / DV) * (1/V) + (D^2)v / DV, where (D^2) is second partial derivative, D is partial derivative, v is radial velocity (ds/dt), c is constant, and radius V = (x^2 + y^2 + z^2)^0.5.
Vibrators with fixed endpoints can have stationary waves. Wave equations {stationary wave equation} can model steady-state waves. Wavefunction del operator, potential energy change, plus constant times wavefunction, kinetic energy change, equals zero {reduced wave equation} {Helmholtz equation}: Dw + (k^2) * w = 0, where w is wavefunction, D is delta function, and k is constant. The solution is an exponential function with complex exponents.
3-Calculus-Differential Equation
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Date Modified: 2022.0225