ODEs Having Linear Symmetries
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Description
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The general forms of ODEs having one of the following linear symmetries
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[xi=a+b*x, eta=0], [xi=a+b*y, eta=0], [xi=0, eta=c+d*x], [xi=0, eta=c+d*y]:
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where the infinitesimal symmetry generator is given by:
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G := f -> xi*diff(f,x) + eta*diff(f,y);
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ode[1] := DEtools[equinv]([xi=a+b*x, eta=0], y(x), 2);
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ode[2] := DEtools[equinv]([xi=a+b*y, eta=0], y(x), 2);
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ode[3] := DEtools[equinv]([xi=0, eta=c+d*x], y(x), 2);
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ode[4] := DEtools[equinv]([xi=0, eta=c+d*y], y(x), 2);
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Although the symmetries of these families of ODEs can be determined in a direct manner (using symgen), the simplicity of their pattern motivated us to have separate routines for recognizing them.
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Examples
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As an example that can be solved by the related routine, consider
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See Also
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DEtools, odeadvisor, dsolve,Lie, and ?odeadvisor,<TYPE> where <TYPE> is one of: quadrature, missing, reducible, linear_ODEs, exact_linear, exact_nonlinear, sym_Fx, linear_sym, Bessel, Painleve, Halm, Gegenbauer, Duffing, ellipsoidal, elliptic, erf, Emden, Jacobi, Hermite, Lagerstrom, Laguerre, Liouville, Lienard, Van_der_Pol, Titchmarsh; for other differential orders see odeadvisor,types.
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