Solving Homogeneous ODEs of Class C
Description
Examples
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≔ⅆⅆxy⁡x=F⁡a⁢x+b⁢y⁡x+cr⁢x+s⁢y⁡x+t
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.
with(DEtools, odeadvisor, symgen);
odeadvisor,symgen
odeadvisor(homogeneousC_ode);
_homogeneous,class C,_dAlembert
A pair of infinitesimals for the homogeneousC_ode
symgen(homogeneousC_ode);
_ξ=a⁢s⁢x−b⁢r⁢x−b⁢t+c⁢sa⁢s−b⁢r,_η=a⁢s⁢y−b⁢r⁢y+a⁢t−c⁢ra⁢s−b⁢r
The general solution for this ODE
ans := dsolve(homogeneousC_ode);
Explicit or implicit results can be tested, in principle, using odetest
odetest(ans,homogeneousC_ode);
0
See Also
DEtools
odeadvisor
dsolve
quadrature
linear
separable
Bernoulli
exact
homogeneous
homogeneousB
homogeneousC
homogeneousD
homogeneousG
Chini
Riccati
Abel
Abel2A
Abel2C
rational
Clairaut
dAlembert
sym_implicit
patterns
odeadvisor,types
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