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Solving Implicit ODEs

Description

  

The sym_implicit subroutine of the odeadvisor command tests if a given first order ODE in "implicit form" (that is, dy/dx cannot be isolated) has one or more of the following symmetries:

[xi=0, eta=y], [xi=0, eta=x], [xi=0, eta=1/x], [xi=0, eta=1/y],
[xi=x, eta=0], [xi=y, eta=0], [xi=1/x, eta=0], [xi=1/y, eta=0],
[xi=x, eta=y], [xi=y, eta=x]:

  

where the infinitesimal symmetry generator is given by the following:

G := f -> xi*diff(f,x) + eta*diff(f,y);

G:=f→ξxf+ηyf

(1)
  

This routine is relevant when using symmetry methods for solving high-degree ODEs (nonlinear in dy/dx). The cases [xi=0, eta=y], [xi=1/x, eta=y] and [xi=x, eta=y] cover the families of homogeneous ODEs mentioned in Murphy's book, pages 63-64.

Examples

withDEtools,odeadvisor

odeadvisor

(2)

Consider the symmetry

X:=1y,0

X:=1y,0

(3)

The most general implicit ODE having this symmetry is given by

implicit_ode:=Fyx,1ⅆⅆxyxx+yxⅆⅆxyx=0

implicit_ode:=Fyx,ⅆⅆxyxx+yxⅆⅆxyx=0

(4)

odeadvisorimplicit_ode

_1st_order,_with_symmetry_[F(x)*G(y),0]

(5)

where F is an arbitrary function of its arguments. Based on this pattern recognition, dsolve solves this ODE as follows

ans:=dsolveimplicit_ode

ans:=yxx∫yxRootOfF_a,_Zⅆ_a_C1=0

(6)

Explicit and implicit answers can be tested, in principle, using odetest

odetestans,implicit_ode

0

(7)

See Also

DEtools, odeadvisor, dsolve, and ?odeadvisor,<TYPE> where <TYPE> is one of: quadrature, linear, separable, Bernoulli, exact, homogeneous, homogeneousB, homogeneousC, homogeneousD, homogeneousG, Chini, Riccati, Abel, Abel2A, Abel2C, rational, Clairaut, dAlembert, sym_implicit, patterns; for other differential orders see odeadvisor,types.


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