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Solving Implicit ODEs
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Description
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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:
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[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]:
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where the infinitesimal symmetry generator is given by the following:
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G := f -> xi*diff(f,x) + eta*diff(f,y);
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This routine is relevant when using symmetry methods for solving high-degree ODEs (non linear 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.
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Examples
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![[odeadvisor]](/support/helpjp/helpview.aspx?si=7219/file04387/math41.png)
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Consider the symmetry
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![X := [1/y, 0]](/support/helpjp/helpview.aspx?si=7219/file04387/math47.png)
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![X := [1/y, 0]](/support/helpjp/helpview.aspx?si=7219/file04387/math50.png)
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The most general implicit ODE having this symmetry is given by
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![[[_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]](/support/helpjp/helpview.aspx?si=7219/file04387/math66.png)
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where F is an arbitrary function of its arguments. Based on this pattern recognition, dsolve solves this ODE as follows
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Explicit and implicit answers can be tested, in principle, using odetest
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See Also
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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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