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

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Differential Equations : Classifying ODEs : First Order : odeadvisor/Abel2C

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

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)

ITR:=yx=1utg1tg0tg1t,x=t

ITR:=x=t,yx=1utg1tg0tg1t

(4)

new_ode:=dchangeITR,Abel_ode2C,ut,t:

new_ode:=collectⅆⅆtut=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, and ?odeadvisor,<TYPE> where <TYPE> is one of: quadrature, linear, separable, Bernoulli, exact, homogeneous, homogeneousB, homogeneousC, homogeneousD, homogeneousG, Chini, Riccati, Abel, Abel2A, rational, Clairaut, dAlembert, sym_implicit, patterns; for other differential orders see odeadvisor,types.


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam