Solving ODEs Matching the Patterns y=G(x,diff(y,x)), x=G(y,diff(y,x)), y=G(diff(y,x)), x=G(diff(y,x)), 0=G(x,diff(y,x)), 0=G(y,diff(y,x)) - Maple Help

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Solving ODEs Matching the Patterns y=G(x,diff(y,x)), x=G(y,diff(y,x)), y=G(diff(y,x)), x=G(diff(y,x)), 0=G(x,diff(y,x)), 0=G(y,diff(y,x))

Description

• 

See Differentialgleichungen, by E. Kamke, p. 30. The technique consists mainly of looking for a parametric solution. Consider, for instance, the case y=G(x,diff(y,x)).

ode := y=G(x,diff(y(x),x));

ode:=y=Gx,ⅆⅆxyx

(1)
  

Choosing the parametrization

para := diff(y(x),x) = t;

para:=ⅆⅆxyx=t

(2)

ode1 := subs(para,x=x(t),y=y(t), ode);

ode1:=yt=Gxt,t

(3)
  

From the equations above and using the chain rule dydx=dydt dtdx, it is possible to obtain another ODE for xt as follows:

ode_draft := diff(x(t),t) = diff(rhs(ode1),t)/t:

ode2 := diff(x(t),t)=solve(ode_draft,diff(x(t),t));

ode2:=ⅆⅆtxt=D2Gxt,tD1Gxt,tt

(4)
  

You should therefore solve ode2 for xt and determine yt by introducing the resulting xt in ode1. Note that, when G does not depend on xt,ode2 is a quadrature. ODEs matching the pattern x=Gy,xy are solved using the same ideas, and ODEs matching the patterns 0=Gx,xy,0=Gy,xy,x=Gxy, or y=Gxy are just particular cases.

• 

Although any ODE can be attempted using the scheme outlined above, generally speaking, there are four cases which can be better dealt with by looking for a parametric solution; they are:

1. 

y=Gx,xy

2. 

x=Gy,xy

3. 

y=Gxy   (particular case)

4. 

x=Gxy   (particular case)

  

Parametric solutions are available by giving the optional argument 'parametric' to dsolve. By default, when the ODE is of high degree in dydx, dsolve tries the parametric scheme, along with a set of related methods for this type of ODE. However, this scheme may also be of help in some cases in which dydx can be isolated.

Examples

1) Kamke's ODE 554: y=G(x,y')

withDEtools,odeadvisor

odeadvisor

(5)

ode:=xn1ⅆⅆxyxnnxⅆⅆxyx+yx

ode:=xn1ⅆⅆxyxnnxⅆⅆxyx+yx

(6)

odeadvisorode

y=_G(x,y')

(7)

dsolveode

yx=_C1x_C11nn1_C1n_C1

(8)

ode:=16yx2ⅆⅆxyx3+2xⅆⅆxyxyx

ode:=16yx2ⅆⅆxyx3+2xⅆⅆxyxyx

(9)

odeadvisorode

_1st_order,_with_linear_symmetries

(10)

dsolveode

yx=1321/431/4x31/4,yx=1321/431/4x31/4,yx=13I21/431/4x31/4,yx=13I21/431/4x31/4,yx=16_C13+2_C1x,yx=16_C13+2_C1x

(11)

3) Kamke's ODE 568: y=Gy'  and d'Alembert type (see odeadvisor,dAlembert)

ode:=ⅆⅆxyx2sinⅆⅆxyxyx

ode:=ⅆⅆxyx2sinⅆⅆxyxyx

(12)

odeadvisorode

_quadrature

(13)

ans:=dsolveode

ans:=yx=0,x∫yx1RootOfsin_Z_Z2_aⅆ_a_C1=0

(14)

Implicit or explicit answers can be tested using odetest; when there are many answers one can map as follows

mapodetest,ans,ode

0,0

(15)

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; for other differential orders see odeadvisor,types.


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