Lagerstrom ODEs - Maple Help

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Lagerstrom ODEs

Description

• 

The general form of the Lagerstrom ODE is given by the following:

Lagerstrom_ode := diff(y(x),x,x)= -k*diff(y(x),x)/x-epsilon*y(x)*diff(y(x),x);

Lagerstrom_ode:=ⅆ2ⅆx2yx=kⅆⅆxyxxεyxⅆⅆxyx

(1)
  

See Rosenblat and Shepherd, "On the Asymptotic Solution of the Lagerstrom Model Equation".

Examples

The second order Lagerstrom ODE can be reduced to a first order ODE of Abel type once the system succeeds in finding one polynomial symmetry for it (see symgen):

withDEtools,odeadvisor,symgen:

odeadvisorLagerstrom_ode

_Lagerstrom,_2nd_order,_with_linear_symmetries

(2)

symgenLagerstrom_ode,way=3

_ξ=x,_η=y

(3)

From which, giving the same indication directly to dsolve, you obtain the reduction of order

ans:=dsolveLagerstrom_ode,way=3

ans:=yx=_aⅇ∫_b_aⅆ_a+_C1 &where ⅆⅆ_a_b_a=_a2ε_ak+2_a_b_a3+_aεk+3_b_a2,_a=xyx,_b_a=1xyx+xⅆⅆxyx,x=1ⅇ∫_b_aⅆ_a+_C1,yx=_aⅇ∫_b_aⅆ_a+_C1

(4)

For the structure of the solution above see ODESolStruc. Reductions of order can also be tested with odetest

odetestans,Lagerstrom_ode

0

(5)

The reduced ODE is of Abel type and can be selected using the mouse, or as follows

reduced_ode:=op2,2,1,1,ans

reduced_ode:=ⅆⅆ_a_b_a=_a2ε_ak+2_a_b_a3+_aεk+3_b_a2

(6)

odeadvisorreduced_ode

_Abel

(7)

See Also

DEtools, odeadvisor, dsolve, and ?odeadvisor,<TYPE> where <TYPE> is one of: quadrature, missing, reducible, linear_ODEs, exact_linear, exact_nonlinear, sym_Fx, linear_sym, Bessel, Painleve, Halm, Gegenbauer, Duffing, ellipsoidal, elliptic, erf, Emden, Jacobi, Hermite, Lagerstrom, Laguerre, Liouville, Lienard, Van_der_Pol, Titchmarsh; for other differential orders see odeadvisor,types.


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