CompleteIntegral
CompleteIntegral[pde,u,{x1,…,xn}]
gives a complete integral u for the first-order partial differential equation pde, with independent variables {x1,…,xn}.
Details and Options
- A complete integral of a first-order partial differential equation (PDE) in n variables is a solution that depends on n independent arbitrary constants c1,c2,…,cn.
- A complete integral is typically used to generate a complete set of solutions to the PDE.
- A solution to the PDE that satisfies a specific initial condition can be obtained by constructing the envelope of a smoothly varying subfamily of simple solutions that depend on
parameters, as illustrated. » - The output from CompleteIntegral is controlled by the form of the dependent function u or u[x1,…,xn], as in DSolve.
- CompleteIntegral can give implicit solutions in terms of Solve.
- CompleteIntegral can give solutions that include Inactive sums and integrals that cannot be carried out explicitly. Variables K[1], K[2], … are used in such cases.
- Boundary conditions for the PDE can be specified to obtain specific solutions of the PDE that are free from the arbitrary constants in the complete integral. »
- The following options can be given:
-
Assumptions $Assumptions assumptions on parameters GeneratedParameters C how to name generated parameters Method Automatic what method to use - GeneratedParameters controls the form of generated parameters; these are by default constants C[n].
Examples
open all close allBasic Examples (3)
Scope (5)
Find the complete integral of a partial differential equation in 2 dimensions:
Get a "pure function" complete integral for u:
Substitute the solution into an expression:
Complete integral that can be expressed using elementary functions:
Complete integral that can be expressed using special functions:
Complete integral for a linear PDE:
Compare with the solution given by DSolve:
Complete integral for a quasi-linear PDE:
Compare with the solution given by DSolve:
Applications (2)
Find a complete integral of the Clairaut equation:
The complete integral is given by a two-parameter family of planes:
Select a one-parameter family of these planes:
Find the envelope of this one-parameter family of planes:
Verify that the envelope is also a solution:
Visualize the one-parameter family of planes and the envelope solution:
Find a complete integral of the Hamilton–Jacobi equation:
Properties & Relations (2)
CompleteIntegral finds a complete integral for a nonlinear PDE:
DSolve returns the same solution with a warning message:
Use CompleteIntegral to find a complete integral for a linear PDE:
DSolve returns the general solution for this PDE:
Text
Wolfram Research (2021), CompleteIntegral, Wolfram Language function, https://reference.wolfram.com/language/ref/CompleteIntegral.html.
CMS
Wolfram Language. 2021. "CompleteIntegral." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CompleteIntegral.html.
APA
Wolfram Language. (2021). CompleteIntegral. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CompleteIntegral.html