Solving Homogeneous ODEs of Class C - Maple Help

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Solving Homogeneous ODEs of Class C

Description

• 

The general form of the homogeneous equation of class C is given by the following:

homogeneousC_ode := diff(y(x),x)=F((a*x+b*y(x)+c)/(r*x+s*y(x)+t));

homogeneousC_ode:=ⅆⅆxyx=Fax+byx+crx+syx+t

(1)
  

where F is an arbitrary function of its argument. See Differentialgleichungen, by E. Kamke, p. 19. This type of ODE can be solved in a general manner by dsolve and the coefficients of the infinitesimal symmetry generator are also found by symgen.

Examples

withDEtools,odeadvisor,symgen

odeadvisor,symgen

(2)

odeadvisorhomogeneousC_ode

_homogeneous,class C,_dAlembert

(3)

A pair of infinitesimals for the homogeneousC_ode

symgenhomogeneousC_ode

_ξ=asxbrxbt+csasbr,_η=asybry+atcrasbr

(4)

The general solution for this ODE

ans:=dsolvehomogeneousC_ode

ans:=yx=atcr+RootOf∫_Z1F_aba_asr+_aⅆ_a+lnxasbrbt+cs+_C1xasbrbt+csas+br

(5)

Explicit or implicit results can be tested, in principle, using odetest

odetestans,homogeneousC_ode

0

(6)

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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