Solving ODEs That Do Not Contain Either the Dependent or Independent Variable - Maple Help

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Solving ODEs That Do Not Contain Either the Dependent or Independent Variable

Description

• 

The general form of an nth order ODE that is missing the dependent variable is:

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

missing_y_ode:=Fx,seqⅆiⅆxiyx,i=1..n

(1)
  

where F is an arbitrary function of its arguments. The order can be reduced by introducing a new variable p(x) = diff(y(x),x). If the reduced ODE can be solved for p(x), the solution to the original ODE is determined as a quadrature.

• 

The general form of an nth order ODE that is missing the independent variable is:

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

missing_x_ode:=Fyx,seqⅆiⅆxiyx,i=1..n

(2)
  

where F is an arbitrary function of its arguments. The transformation y' =p,y''=pp',y'''=p2p''+pp'2,...

  

yields a reduction of order. If the reduced ODE can be solved for p(y), the solution to the original ODE can be given implicitly as

x = Int(1/p(y),y) + _C1;

x=∫1pyⅆy+_C1

(3)
  

See Murphy, "Ordinary Differential Equations and their Solutions", 1960, sections B2(1,2), and C2(1,2).

Examples

withDEtools,odeadvisor

odeadvisor

(4)

_2nd_order_missing_x_ode:=ⅆ2ⅆx2yx=lnyx+1ⅆⅆxyx

_2nd_order_missing_x_ode:=ⅆ2ⅆx2yx=lnyx+1ⅆⅆxyx

(5)

odeadvisor_2nd_order_missing_x_ode

_2nd_order,_missing_x,_2nd_order,_exact,_nonlinear,_2nd_order,_reducible,_mu_x_y1,_2nd_order,_reducible,_mu_xy

(6)

solx2:=dsolve_2nd_order_missing_x_ode

solx2:=yx=_C1,∫yx1_aln_a+_C1ⅆ_ax_C2=0

(7)

Explicit and implicit answers can be tested, in principle, using odetest:

mapodetest,solx2,_2nd_order_missing_x_ode

0,0

(8)

_2nd_order_missing_y_ode:=ⅆ2ⅆx2yx=Fxⅆⅆxyx3

_2nd_order_missing_y_ode:=ⅆ2ⅆx2yx=Fxⅆⅆxyx3

(9)

odeadvisor_2nd_order_missing_y_ode

_2nd_order,_missing_y,_2nd_order,_reducible,_mu_y_y1

(10)

soly2:=dsolve_2nd_order_missing_y_ode

soly2:=yx=∫1_C12∫Fxⅆxⅆx+_C2,yx=∫1_C12∫Fxⅆxⅆx+_C2

(11)

In the case of multiple answers it is convenient to "map" odetest as follows:

mapodetest,soly2,_2nd_order_missing_y_ode

0,0

(12)

The most general third order ODE missing x. This ODE cannot be solved to the end: its solution involves the solving of the most general second order ODE. However, its differential order can be reduced (see ?dsolve,ODESolStruc):

_3rd_order_missing_x_ode:=ⅆ3ⅆx3yx=Fyx,ⅆⅆxyx,ⅆ2ⅆx2yx

_3rd_order_missing_x_ode:=ⅆ3ⅆx3yx=Fyx,ⅆⅆxyx,ⅆ2ⅆx2yx

(13)

solx3:=dsolve_3rd_order_missing_x_ode

solx3:=yx=_a &where ⅆ2ⅆ_a2_b_a_b_a2+ⅆⅆ_a_b_a2_b_aF_a,_b_a,ⅆⅆ_a_b_a_b_a=0,_a=yx,_b_a=ⅆⅆxyx,x=∫1_b_aⅆ_a+_C1,yx=_a

(14)

odeadvisor_3rd_order_missing_x_ode

_3rd_order,_missing_x,_3rd_order,_with_linear_symmetries

(15)

odetestsolx3,_3rd_order_missing_x_ode

0

(16)

The most general third order ODE missing y.

_3rd_order_missing_y_ode:=ⅆ3ⅆx3yx=Fx,ⅆⅆxyx,ⅆ2ⅆx2yx

_3rd_order_missing_y_ode:=ⅆ3ⅆx3yx=Fx,ⅆⅆxyx,ⅆ2ⅆx2yx

(17)

odeadvisor_3rd_order_missing_y_ode

_3rd_order,_missing_y,_3rd_order,_with_linear_symmetries

(18)

soly3:=dsolve_3rd_order_missing_y_ode

soly3:=yx=∫_b_aⅆ_a+_C1 &where ⅆ2ⅆ_a2_b_a=F_a,_b_a,ⅆⅆ_a_b_a,_a=x,_b_a=ⅆⅆxyx,x=_a,yx=∫_b_aⅆ_a+_C1

(19)

odetestsoly3,_3rd_order_missing_y_ode

0

(20)

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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