Solving Abel's ODEs of the Second Kind, Class C - Maple Programming Help

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Solving Abel's ODEs of the Second Kind, Class C

 

Description

Examples

Description

• 

The general form of Abel's equation, second kind, class C is given by:

Abel_ode2C := (g1(x)*y(x)+g0(x))*diff(y(x),x)
= f3(x)*y(x)^3 + f2(x)*y(x)^2 + f1(x)*y(x) + f0(x);

Abel_ode2C:=g1xyx+g0xⅆⅆxyx=f3xyx3+f2xyx2+f1xyx+f0x

(1)
  

where f3(x), f2(x), f1(x), f0(x), g1(x) and g0(x) are arbitrary functions. See Differentialgleichungen, by E. Kamke, p. 28. There is as yet no general solution for this ODE.

Examples

withDEtools,odeadvisor

odeadvisor

(2)

All ODEs of type Abel, second kind, can be rewritten as ODEs of type Abel, first kind, using the following transformation:

withPDEtools,dchange

dchange

(3)

ITRyx=1utg1tg0tg1t,x=t

ITR:=x=t,yx=1utg1tg0tg1t

(4)

new_odedchangeITR,Abel_ode2C,ut,t:

new_odecollectⅆⅆtut=solvenew_ode,ⅆⅆtut,ut

new_ode:=ⅆⅆtut=f3tg0t3g1t3f0t+g1t2g0tf1tg1tg0t2f2tut3g1t2+3f3tg0t2g1t2f1tg1t2ⅆⅆtg0t+2g1tg0tf2t+g1tg0tⅆⅆtg1tut2g1t2+3f3tg0tg1tf2tg1tⅆⅆtg1tutg1t2f3tg1t2

(5)

odeadvisornew_ode,ut,Abel

_Abel

(6)

See Also

DEtools

odeadvisor

dsolve

quadrature

linear

separable

Bernoulli

exact

homogeneous

homogeneousB

homogeneousC

homogeneousD

homogeneousG

Chini

Riccati

Abel

Abel2A

rational

Clairaut

dAlembert

sym_implicit

patterns

odeadvisor,types

 


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