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Calculus`DSolveIntegrals`

Most nonlinear partial differential equations do not allow for general solution, in which case it is advisable to use the CompleteIntegral function that attempts to find a sufficiently representative family of particular solutions called a complete integral.
A complete integral of an equation is known to be exhaustive in the sense that solutions of almost all boundary value problems for the equation may then be expressed in quadratures of the complete integral. Thus, the complete integral plays a role similar to that of the Green's function for linear second-order partial differential equations.


Finding the complete integral of an equation.

  • This loads the package.
  • In[1]:= <<Calculus`DSolveIntegrals`

  • Here B[1] and B[2] are parameters of the solution.
  • In[2]:= CompleteIntegral[
    Derivative[0, 1][u][x, y] == (u[x, y] +
    x^2*Derivative[1, 0][u][x, y]^2)/y,
    u[x,y], {x,y}]

    Out[2]=

    B[n] is the default name for the parameters appearing in the result of CompleteIntegral. The names of parameters may be selected using the option IntegralConstants just as the names of the undetermined constants in the result of DSolve may be selected using the option DSolveConstants.

  • This shows how the names of parameters of the complete integral may be changed. The integral is Example 6.7 from KamkeII.
  • In[3]:= CompleteIntegral[-u[x, y] +
    (2 + y)*Derivative[0, 1][u][x, y] +
    x*Derivative[1, 0][u][x, y] +
    3*Derivative[1, 0][u][x, y]^2 == 0,
    u[x,y], {x,y}, IntegralConstants->F]

    Out[3]=

    For the needs of advanced users (especially those dealing with analytical mechanics), a function is supported by the package for finding the differential invariants ("first integrals", "constants of motion") of systems of ordinary differential equations.


    Finding the differential invariants for a system of differential equations.

  • This builds a list of two independent invariants for a two-dimensional differential system. The dependence of and on

    is suppressed.
  • In[4]:= DifferentialInvariants[
    {u'[x] == -(u[x] (u[x] + v[x])),
    v'[x] == v[x] (u[x] + v[x])},
    {u, v}, x]

    Out[4]=

    Reference

    [KamkeII] E. Kamke, Differentialgleichungen, Lösungsmethoden und Lösungen, Band 2, Partielle Differentialgleichungen Erster Ordnung fur Eine Gesuchte Funktion, Academische Verlagsgesellschaft, Leipzig, 1948.