# 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, K, 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 allclose all

## Basic Examples(3)

Find a complete integral of a partial differential equation:

Plot the solution for particular values of the arbitrary constants:

Get a "pure function" complete integral for w:

Substitute the solution into an expression:

Complete integral of a partial differential equation in 3 dimensions:

## 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 HamiltonJacobi equation:

Compute an envelope solution for the equation:

Plot the envelope solution:

Verify that the envelope solution satisfies the 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: