DSolveChangeVariables

DSolveChangeVariables[dsolve,u,t,trans]

changes the solution function in dsolve to using the transformation trans.

DSolveChangeVariables[dsolve,{u1,u2,…},t,trans]

changes the solution functions in the system to .

DSolveChangeVariables[dsolve,u,{t1,…,tn},trans]

changes the solution function in the partial differential equation to .

Details and Options

  • A change of variables is often used to simplify the coefficients in a differential expression or to represent it in a more suitable coordinate system such as polar coordinates to exploit the symmetry in the problem.
  • DSolveChangeVariables can be used to perform a change of variables for a single ordinary differential equation or partial differential equation without initial or boundary conditions.
  • The change of variables is performed using the chain rule
  • on an interval or
  • over a region where denotes the Jacobian of function with respect to its arguments.
  • The possible forms for dsolve are the forms supported by DSolve:
  • DSolve[deq,y,x]ordinary differential equation
    DSolve[{deq1,…,deqn},{y1,…,yn},x]system of differential equations
    DSolve[deq,z,{x,y,…}]partial differential equation
  • Either an unevaluated DSolve[…] or Inactive[DSolve][…] can be used. It is important that the dsolve does not evaluate, so the safe method is to use Inactive[DSolve][…], which can be produced through Inactivate[dsolve,DSolve].
  • DSolveChangeVariables returns the result in the form Inactive[DSolve][…]. Use Activate to solve the differential equation in the new coordinates. »
  • The transformation trans can have the forms:
  • t==ϕ[x]replace ϕ[x] by t
    {u==ϕ[x,y,…],v=ψ[x,y,…],…}replace ϕ[x,y,…] by u and ψ[x,y,…] by v, etc.
    chart1chart2named coordinate systems from CoordinateChartData
  • The transformation is assumed to be differentiable on its domain of definition.
  • When using named coordinate systems, the transformation can be entered in any form accepted by CoordinateTransformData, including {oldsys,metric,dim}{newsys,metric,dim}, {oldsysnewsys,metric,dim} and the various more abbreviated forms.
  • Restrictions on the domains of the variables and parameters in the differential expression can be specified using Assumptions.

Examples

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Basic Examples  (4)

Apply the change of variables to a linear differential equation:

Solve the differential equation:

Apply the change of variables to a nonlinear differential equation:

Transform the 2D Laplace equation from Cartesian to the polar coordinate system:

Apply the change of variables to a linear ordinary differential equation:

Solve the resulting differential equation and substitute back to the original variable:

Compare the result with the solution to the original ordinary differential equation:

Scope  (18)

Ordinary Differential Equations  (6)

Simplify a second-order linear differential equation with a change of variables:

Transform the nonlinear Riccati equation to a linear ODE with the change of variables :

Apply the trigonometric transformation to an ordinary differential equation:

Apply the complex-valued transformation and to an ordinary differential equation:

Apply the change of variables to a differential equation:

Reduce a variable-coefficient nonlinear ODE to a constant-coefficient equation using :

Partial Differential Equations  (5)

Transform the -dimensional wave equation into null coordinates:

Apply the same transformation but express the old coordinates in terms of the new ones:

Apply the change of variables to the heat equation:

Apply the change of variables and to a partial differential equation:

Apply the change of variables and to a partial differential equation:

Make a purely symbolic transformation to a partial differential equation:

Partial Differential Equations and Named Coordinate Systems  (7)

Transform a 3D Poisson's equation from Cartesian to the cylindrical coordinate system:

Transform the 3D wave equation from Cartesian to the cylindrical coordinate system:

Transform the 3D biharmonic equation from Cartesian to the spherical coordinate system:

Transform the 3D heat equation in spherical coordinates back to the Cartesian coordinates:

Transform a three-dimensional Schrödinger in spherical coordinates to cylindrical coordinates:

Express the 4D Laplace equation in hyperspherical coordinates:

Transform Poisson's equation on the sphere from standard angular to stereographic coordinates:

Applications  (6)

In quantum mechanics, the operator is a multiple of the angular momentum in the direction. Show this by transforming the equation to polar coordinates:

Consider the Cauchy–Euler equation . It is possible to transform this ODE into an equation with constant coefficients by applying the change of variables :

Apply the transformation to the linear second-order ordinary differential equation :

Dividing the transformed equation by leads to a drastically simplified differential equation:

A spherically symmetric solution to the heat equation (partialf)/(partialt)=TemplateBox[{f, }, Laplacian] in three dimensions can be reduced to a linear ODE. First, write out the equation in spherical coordinates:

Then change to a function of a single variable that combines both space and time coordinates:

In the heat-mass transfer problem, an equation takes the form TemplateBox[{H, }, Laplacian]=u (((1-mu^2) )/r(partialH(r,mu))/(partialmu)+mu (partialH(r,mu))/(partialr)), where is a constant, is the distance from the origin, and is a modification of the polar angle. Starting from standard spherical coordinates, express this equation in local coordinates, then show it can be reduced to Poisson's equation. Set TemplateBox[{H, }, Laplacian] equal to a constant and use DSolveChangeVariables to change how the polar angle is expressed:

Create an operator to express the Laplacian in these coordinates:

Change to new independent variable using :

Divide out the exponential and isolate the term free of the derivatives on the right-hand side:

The left-hand side is the Laplacian in modified spherical coordinates, confirming this is Poisson's equation:

Apply the separation of variables transformation to a Heat equation–type partial differential equation:

After dividing by , you can see that the variables are separated:

Properties & Relations  (2)

A result always has an inactive head, irrespective of the form of input:

DSolveChangeVariables effectively uses the "Mapping" property of CoordinateTransformData:

Possible Issues  (1)

DSolveChangeVariables transforms differential operators, here giving :

Linear differential operators can be transformed as vector fields, though TransformedField will express the result in terms of orthonormal bases, here giving :

Wolfram Research (2022), DSolveChangeVariables, Wolfram Language function, https://reference.wolfram.com/language/ref/DSolveChangeVariables.html.

Text

Wolfram Research (2022), DSolveChangeVariables, Wolfram Language function, https://reference.wolfram.com/language/ref/DSolveChangeVariables.html.

CMS

Wolfram Language. 2022. "DSolveChangeVariables." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DSolveChangeVariables.html.

APA

Wolfram Language. (2022). DSolveChangeVariables. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DSolveChangeVariables.html

BibTeX

@misc{reference.wolfram_2024_dsolvechangevariables, author="Wolfram Research", title="{DSolveChangeVariables}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/DSolveChangeVariables.html}", note=[Accessed: 22-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_dsolvechangevariables, organization={Wolfram Research}, title={DSolveChangeVariables}, year={2022}, url={https://reference.wolfram.com/language/ref/DSolveChangeVariables.html}, note=[Accessed: 22-November-2024 ]}