Solving ODEs of the Form Chini - Maple Help

Online Help

All Products    Maple    MapleSim


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

Solving ODEs of the Form Chini

 

Description

Examples

Description

• 

The general form of Chini's equation is given by:

Chini_ode := diff(y(x),x)=f(x)*y(x)^n-g(x)*y(x)+h(x);

Chini_ode:=ⅆⅆxyx=fxyxngxyx+hx

(1)
  

where f(x), g(x) and h(x) are arbitrary functions. See Differentialgleichungen, by E. Kamke, ODE 1.55, p. 303.

  

There is as yet no general solution for this ODE. For n=2, Chini's ODE is of Riccati type; for n=3 it is of Abel type. If the following combination of coefficients

Chini_invariant := f(x)^(-n-1)*h(x)^(-2*n+1)*(f(x)*
diff(h(x),x)-diff(f(x),x)*h(x)-g(x)*f(x)*n*h(x))^n*n^(-n);

Chini_invariant:=fxn1hx2n+1fxⅆⅆxhxⅆⅆxfxhxgxfxnhxnnn

(2)
  

is independent of 'x', then the solution to the ODE follows in a straightforward manner; see Kamke, page 303. This scheme, proposed by Chini, generalizes the method of invariants for Abel ODEs (also proposed by Chini) found in Kamke's book as sub-method (g) for Abel ODEs; see odeadvisor,Abel.

  

Actually, when the square term in the Abel ODE is zero, the Abel invariant described in odeadvisor,Abel is equal to the Chini invariant described below. Now all Abel ODEs can be rewritten in Chini format (that is, square term = 0; see example below), and if the Abel invariant of an Abel ODE is constant, then the Abel invariant for this ODE written in Chini format is also constant. To understand this, note that the "independence of the invariant with respect to 'x'" for Abel ODEs is preserved under transformations of the form

y -> G(t)*u(t)+H(t);

y→Gtut+Ht

(3)

x -> F(t);

x→Ft

(4)
  

for any G, H, and F. The transformation that removes the square term in Abel ODEs (that is, rewrites it in Chini format) is given by

y -> u(x)-f2/(f3*3);

y→ux13f2f3

(5)
  

and is just a particular case of the general transformation displayed above (see also odeadvisor,Abel.

Examples

withPDEtools,dchange

dchange

(6)

withDEtools,odeadvisor

odeadvisor

(7)

odeadvisorChini_ode

_Chini

(8)

All Abel ODEs, such as the following:

Abel_odeⅆⅆxyx=f3xyx3+f2xyx2+f1xyx+f0x

Abel_ode:=ⅆⅆxyx=f3xyx3+f2xyx2+f1xyx+f0x

(9)

can be written in 'Chini' form by eliminating the square term through the following transformation:

tryx=uxf2x3f3x

tr:=yx=ux13f2xf3x

(10)

Abel_ode_Chinicollectⅆⅆxux=solvedchangetr,Abel_ode,u,ⅆⅆxux,ux

Abel_ode_Chini:=ⅆⅆxux=f3xux3+12727f3x2f1x9f3xf2x2uxf3x2+12727f0xf3x29f3xf2xf1x+2f2x3+9ⅆⅆxf2xf3x9f2xⅆⅆxf3xf3x2

(11)

coeffrhsAbel_ode_Chini,ux,2

0

(12)

Kamke, 1.52:

odeⅆⅆxyx=ayxn+bxn1n

ode:=ⅆⅆxyx=ayxn+bxnn+1

(13)

odeadvisorode

_homogeneous,class G,_Chini

(14)

ansdsolveode,yx

ans:=∫_byxxnn1axn1_an+_axnn1+bn1xⅆ_an1+lnx_C1=0

(15)

Kamke, 1.54:

odeⅆⅆxyxanfx1nⅆⅆxgxyxnⅆⅆxfxyxfxfxⅆⅆxgx

ode:=ⅆⅆxyxanfxn+1ⅆⅆxgxyxnⅆⅆxfxyxfxfxⅆⅆxgx

(16)

odeadvisorode,yx

_Chini,_1st_order,_with_symmetry_[F(x),G(x)*y+H(x)]

(17)

dsolveode,yx

ayxLerchPhiayxfxn,1,1nnfxagx+_C1=0

(18)

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

 


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