Exact Nonlinear ODEs - Maple Programming Help

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Exact Nonlinear ODEs

 

Description

Examples

Description

• 

The general form of the exact nonlinear ODE is given by the following:

exact_nonlinear_ode := 'diff(F(x,y(x),seq(diff(y(x),x$i),i=1..n)),x)' = 0;

exact_nonlinear_ode:=xFx,yx,seqⅆiⅆxiyx,i=1..n=0

(1)
  

See Murphy, "Ordinary Differential Equations and their Solutions", p. 221.

• 

The order of this ODE can be reduced since it is the total derivative of an ODE of one order lower. If the given ODE is G(x,y,y1,y2,...,yn)=0, the test for exactness is the following:

g0Dg1+D2g2±Dngn=0

  

where

D=ⅆⅆx,y1,,ynbeingfunctionsofx

gn=Dn+1Gx,y,y1,y2,...,yn=ⅆGⅆyn,

yn=ⅆnⅆxnyx

  

Note: The derivatives with respect to y, dy/dx and d^2y/dx^2 are taken in the obvious manner but the derivatives with regard to x are taken considering y, and its derivatives as functions of x.

  

The reduced ODE is:

reduced_ode := 'F(x,y(x),seq(diff(y(x),x$i),i=1..n))' = _C1;

reduced_ode:=Fx,yx,seqⅆiⅆxiyx,i=1..n=_C1

(2)

Examples

withDEtools,odeadvisor

odeadvisor

(3)

odeⅆ2ⅆx2yx=1yxxⅆⅆxyxyx2

ode:=ⅆ2ⅆx2yx=1yxxⅆⅆxyxyx2

(4)

odeadvisorode

_2nd_order,_exact,_nonlinear,_2nd_order,_with_linear_symmetries,_2nd_order,_reducible,_mu_x_y1,_2nd_order,_reducible,_mu_y_y1,_2nd_order,_reducible,_mu_xy

(5)

ansdsolveode,implicit

ans:=12ln_C1xyxx2+yx2x2_C1arctanh_C1x+2yxx_C12+4_C12+4lnx_C2=0

(6)

odetestans,ode

0

(7)

See Also

DEtools

odeadvisor

dsolve

quadrature

missing

reducible

linear_ODEs

exact_linear

exact_nonlinear

odeadvisor,types

 


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