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

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

 

Description

Examples

Description

• 

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

Abel_ode2A := (y(x)+g(x))*diff(y(x),x)=f2(x)*y(x)^2+f1(x)*y(x)+f0(x);

Abel_ode2A:=yx+gxⅆⅆxyx=f2xyx2+f1xyx+f0x

(1)
  

where f2(x), f1(x), f0(x), and g(x) are arbitrary functions. See Differentialgleichungen, by E. Kamke, p. 26. There is as yet no general solution for this ODE.

• 

Note that all ODEs of type Abel, second kind, can be rewritten as ODEs of type Abel, first kind, as explained in ?odeadvisor,Abel2C

Examples

withDEtools,symgen,odeadvisor

symgen,odeadvisor

(2)

odeadvisorAbel_ode2A

_Abel,2nd type,class A

(3)

1) f0(x) = f1(x)*g(x)-f2(x)*g(x)^2

odeevalsubsf0x=f1xgxf2xgx2,Abel_ode2A

ode:=yx+gxⅆⅆxyx=f2xyx2+f1xyx+f1xgxf2xgx2

(4)

This case can be solved as follows:

dsolveode,yx

yx=gx,yx=∫f2xgx+f1xⅇ∫f2xⅆxⅆx+_C1ⅇ∫f2xⅆx

(5)

2) Another case which can be solved:

f1(x) = 2*f2(x)*g(x)-diff(g(x),x)

odeevalsubsf1x=2f2xgxⅆⅆxgx,Abel_ode2A

ode:=yx+gxⅆⅆxyx=f2xyx2+2f2xgxⅆⅆxgxyx+f0x

(6)

Although the answer for this case can be obtained using standard methods (an integrating factor is easily found), the use of symmetry methods can provide an explicit solution. The infinitesimals for this case are given by

symgenode,yx

_ξ=0,_η=ⅇ∫2f2xⅆxy+gx

(7)

To indicate the use of symmetry methods "at first", we can explicitly indicate an integration method (see dsolve); for instance, to use the canonical coordinates of the invariance group:

ansdsolveode,yx,can

ans:=yx=ⅇ2∫f2xⅆxgxⅇ2∫f2xⅆx2gx2+2ⅇ2∫f2xⅆx∫f0xⅇ∫f2xⅆx2ⅆx+2ⅇ2∫f2xⅆx_C1ⅇ2∫f2xⅆx,yx=ⅇ2∫f2xⅆxgx+ⅇ2∫f2xⅆx2gx2+2ⅇ2∫f2xⅆx∫f0xⅇ∫f2xⅆx2ⅆx+2ⅇ2∫f2xⅆx_C1ⅇ2∫f2xⅆx

(8)

See Also

DEtools

dsolve

odeadvisor

quadrature

linear

separable

Bernoulli

exact

homogeneous

homogeneousB

homogeneousC

homogeneousD

homogeneousG

Chini

Riccati

Abel

Abel2C

rational

Clairaut

dAlembert

sym_implicit

patterns

odeadvisor,TYPES

 


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