Solving Homogeneous ODEs of Class D - Maple Help

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Solving Homogeneous ODEs of Class D

Description

• 

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

homogeneousD_ode := diff(y(x),x)= y(x)/x+g(x)*f(y(x)/x);

homogeneousD_ode:=ⅆⅆxyx=yxx+gxfyxx

(1)
  

where f(y(x)/x) and g(x) are arbitrary functions of their arguments. See Differentialgleichungen, by E. Kamke, p. 20. 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,symtest

odeadvisor,symgen,symtest

(2)

odeadvisorhomogeneousD_ode

_homogeneous,class D

(3)

A pair of infinitesimals for homogeneousD_ode

symgenhomogeneousD_ode

_ξ=xgx,_η=ygx

(4)

The general solution for this ODE

ans:=dsolvehomogeneousD_ode

ans:=yx=RootOf∫_Z1f_aⅆ_a+∫gxxⅆx+_C1x

(5)

Answers can be tested using odetest

odetestans,homogeneousD_ode

0

(6)

Let's see how the answer above works when turning f into an explicit function; f is the identity mapping.

f:=u→u

f:=u→u

(7)

allvaluesvalueans

yx=ⅇ∫gxxⅆx+_C1x

(8)

odetestans,homogeneousD_ode

0

(9)

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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