Solving Second and Third Order ODEs using an Integrating Factor - Maple Help

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Solving Second and Third Order ODEs using an Integrating Factor

Description

• 

If, for an nth order ODE (n=2 or n=3) with the nth derivative isolated, there exists an integrating factor which depends only on the (n-1)st derivative, this integrating factor can be determined. The differential order of the ODE can then be reduced by one.

• 

The general form of such an ODE of second order is:

reducible_ode_2 :=
diff(y(x),x,x)=diff(G(x,y(x)),x)/D(F)(diff(y(x),x));

reducible_ode_2:=ⅆ2ⅆx2yx=D1Gx,yx+D2Gx,yxⅆⅆxyxDFⅆⅆxyx

(1)
  

where F and G are arbitrary functions of their arguments. The integrating factor in this case is

mu := D(F)(diff(y(x),x));

μ:=DFⅆⅆxyx

(2)
  

The reduced ODE then becomes

F(diff(y(x),x)) = G(x,y(x)) + _C1;

Fⅆⅆxyx=Gx,yx+_C1

(3)
• 

The general form of this ODE of third order is:

reducible_ode_3 :=
diff(y(x),x$3)=diff(G(x,y(x),diff(y(x),x)),x)/D(F)(diff(y(x),x,x));

reducible_ode_3:=ⅆ3ⅆx3yx=D1Gx,yx,ⅆⅆxyx+D2Gx,yx,ⅆⅆxyxⅆⅆxyx+D3Gx,yx,ⅆⅆxyxⅆ2ⅆx2yxDFⅆ2ⅆx2yx

(4)
  

where F and G are arbitrary functions of their arguments. The integrating factor in this case is

mu := D(F)(diff(y(x),x,x));

μ:=DFⅆ2ⅆx2yx

(5)
  

The reduced ODE is

F(diff(y(x),x,x)) = G(x,y(x),diff(y(x),x)) + _C1;

Fⅆ2ⅆx2yx=Gx,yx,ⅆⅆxyx+_C1

(6)

Examples

withDEtools,odeadvisor

odeadvisor

(7)

ode1:=xⅆ2ⅆx2yx+2ⅆⅆxyxx2xⅆ2ⅆx2yx+2ⅆⅆxyxx2yx2+2ⅆⅆxyx2xyx3=0

ode1:=xⅆ2ⅆx2yx+2ⅆⅆxyxx2xⅆ2ⅆx2yx+2ⅆⅆxyxx2yx2+2ⅆⅆxyx2xyx3=0

(8)

odeadvisorode1

_Liouville,_2nd_order,_with_linear_symmetries,_2nd_order,_reducible,_mu_x_y1,_2nd_order,_reducible,_mu_xy

(9)

sol:=dsolveode1

sol:=yx=12_C2x_C1+_C22x22_C1_C2x+_C124x2x,yx=12_C2x+_C22x22_C1_C2x+_C124x2+_C1x

(10)

Explicit or implicit results can be tested, in principle, using odetest. When testing multiple solutions, you can use map, as follows:

mapodetest,sol,ode1

0,0

(11)

A third order ODE

ode2:=1ⅆ3ⅆx3yxxⅆ2ⅆx2yx=1ⅆⅆxyxx+yxx2yx

ode2:=ⅆ3ⅆx3yxxⅆ2ⅆx2yx=ⅆⅆxyxx+yxx2yx

(12)

odeadvisorode2

_3rd_order,_with_linear_symmetries,_3rd_order,_reducible,_mu_y2,_3rd_order,_reducible,_mu_poly_yn

(13)

sol:=dsolveode2

sol:=yx=_C2AiryAi_C1x+_C3AiryBi_C1x

(14)

See Also

DEtools, odeadvisor, dsolve, and ?odeadvisor,<TYPE> where <TYPE> is one of: quadrature, missing, reducible, linear_ODEs, exact_linear, exact_nonlinear; for other differential orders see odeadvisor,types.


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