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Solving Exact Linear ODEs

Description

• 

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

exact_linear_ode := diff(linear_ODE(x),x) = 0;

exact_linear_ode:=ⅆⅆxlinear_ODEx=0

(1)
  

where linearODE(x) is a linear ODE of any differential order; see Murphy, "Ordinary Differential Equations and their Solutions", p. 221. The order of these exact linear ODEs can be reduced since they are the total derivative of an ODE of one order lower. The reduced ODE is:

linear_ODE(x) + _C1;

linear_ODEx+_C1

(2)

Examples

The most general exact linear non-homogeneous ODE of second order; this case is solvable.

withDEtools,odeadvisor

odeadvisor

(3)

ODE:=ⅆⅆxⅆⅆxyx=Axyx+Bx

ODE:=ⅆ2ⅆx2yx=ⅆⅆxAxyx+Axⅆⅆxyx+ⅆⅆxBx

(4)

odeadvisorODE,yx

_2nd_order,_exact,_linear,_nonhomogeneous

(5)

dsolveODE,yx

yx=_C2+∫_C1+Bxⅇ∫Axⅆxⅆxⅇ∫Axⅆx

(6)

The general exact linear ODE of fourth order which can be reduced to an exact linear ODE of third order; this can be reduced to a second order ODE and the answer is expressed using DESol

ODE:=ⅆ2ⅆx2ⅆ2ⅆx2yx=Axyx+Bxⅆⅆxyx+Fx

ODE:=ⅆ4ⅆx4yx=ⅆ2ⅆx2Axyx+2ⅆⅆxAxⅆⅆxyx+Axⅆ2ⅆx2yx+ⅆ2ⅆx2Bxⅆⅆxyx+2ⅆⅆxBxⅆ2ⅆx2yx+Bxⅆ3ⅆx3yx+ⅆ2ⅆx2Fx

(7)

odeadvisorODE,yx

_high_order,_exact,_linear,_nonhomogeneous

(8)

dsolveODE,yx

yx=DESolAx_YxBxⅆⅆx_Yx+ⅆ2ⅆx2_Yx_C2x_C1Fx,_Yx

(9)

The general exact linear ODE of fifth order which can be reduced to a first order linear ODE. This ODE can be solved to the end.

ODE:=ⅆ4ⅆx4ⅆⅆxyx=Axyx+Bx

ODE:=ⅆ5ⅆx5yx=ⅆ4ⅆx4Axyx+4ⅆ3ⅆx3Axⅆⅆxyx+6ⅆ2ⅆx2Axⅆ2ⅆx2yx+4ⅆⅆxAxⅆ3ⅆx3yx+Axⅆ4ⅆx4yx+ⅆ4ⅆx4Bx

(10)

odeadvisorODE,yx

_high_order,_fully,_exact,_linear

(11)

ans:=dsolveODE,yx

ans:=yx=_C5+∫4_C1x3+3_C2x2+2_C3x+_C4+Bxⅇ∫Axⅆxⅆxⅇ∫Axⅆx

(12)

odetestans,ODE

0

(13)

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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