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Solving Linear ODEs

Description

• 

The general form of the linear ODE is given by:

linear_ODE:=A0yx+A1ⅆⅆxyx+A2ⅆ2ⅆx2yx+...=Fx

  

where the coefficients An can be functions of x, see Differentialgleichungen, by E. Kamke, p. 69. Roughly speaking, there is no general method for solving the most general linear ODE of differential order greater than one. However, this is an active research area and there are many solving schemes which are applicable when the linear ODE satisfies certain conditions. In all the cases, if the method is applicable and the ODE is of second order, the ODE can be integrated to the end; otherwise, its order can be reduced by one or more, depending on the case. A summary of the methods implemented in dsolve for linear ODEs is as follows:

– 

the ODE is exact (see odeadvisor, exact_linear);

– 

the coefficients An are rational functions and the ODE has exponential solutions (see DEtools, expsols);

– 

the ODE has liouvillian solutions (see DEtools, kovacicsols);

– 

the ODE has three regular singular points (see DEtools, RiemannPsols).

– 

the ODE has simple symmetries of the form 0,Fx (see odeadvisor, sym_Fx);

– 

the ODE has special functions" solutions (see odeadvisor, classifications for second order ODEs).

Examples

The most general exact linear non-homogeneous ODE of second order; this case is solvable (see odeadvisor, exact_linear):

withDEtools,odeadvisor

odeadvisor

(1)

ode1:=ⅆⅆxⅆⅆxyx=Axyx+Bx

ode1:=ⅆ2ⅆx2yx=ⅆⅆxAxyx+Axⅆⅆxyx+ⅆⅆxBx

(2)

odeadvisorode1,yx

_2nd_order,_exact,_linear,_nonhomogeneous

(3)

dsolveode1,yx

yx=_C2+∫_C1+Bxⅇ∫Axⅆxⅆxⅇ∫Axⅆx

(4)

Exponential solutions for a third order linear ODE .

ode2:=x2+xⅆ3ⅆx3yxx22ⅆ2ⅆx2yxx+2ⅆⅆxyx=0

ode2:=x2+xⅆ3ⅆx3yxx22ⅆ2ⅆx2yxx+2ⅆⅆxyx=0

(5)

dsolveode2

yx=_C1+_C2lnx+_C3ⅇx

(6)

An example of an ODE with regular singular points

ode3:=x1xⅆ2ⅆx2yx+ca+b+1xⅆⅆxyxabyx

ode3:=x1xⅆ2ⅆx2yx+ca+b+1xⅆⅆxyxabyx

(7)

dsolveode3

yx=_C1hypergeoma,b,c,x+_C2x1chypergeombc+1,ac+1,2c,x

(8)

An example for which symmetries of the form 0,Fx can be found (see odeadvisor, sym_Fx)

ode4:=ⅆ2ⅆx2yx=lnxⅆⅆxyx+yx1+lnx

ode4:=ⅆ2ⅆx2yx=lnxⅆⅆxyx+yx1+lnx

(9)

odeadvisorode4

_2nd_order,_with_linear_symmetries,_2nd_order,_linear,_with_symmetry_[0,F(x)]

(10)

dsolveode4

yx=∫xxⅇxⅇx2ⅆx_C1+_C2ⅇx

(11)

Some ODEs with special function solutions (see odeadvisor, second order ODEs).

Bessel ODE.

ode5:=x2ⅆ2ⅆx2yx+xⅆⅆxyx+x2n2yx=0

ode5:=ⅆ2ⅆx2yxx2+ⅆⅆxyxx+n2+x2yx=0

(12)

odeadvisorode5

_Bessel

(13)

dsolveode5

yx=_C1BesselJn,x+_C2BesselYn,x

(14)

Complete Elliptic Integral ODE.

ode6:=ⅆⅆxx1x2ⅆⅆxyxxyx=0

ode6:=x2+1ⅆⅆxyx2x2ⅆⅆxyx+xx2+1ⅆ2ⅆx2yxxyx=0

(15)

odeadvisorode6

_elliptic,_class_I

(16)

dsolveode6

yx=_C1EllipticKx+_C2EllipticCKx

(17)

Gegenbauer ODE.

ode7:=x21ⅆⅆxⅆⅆxyx2m+3xⅆⅆxyx+λyx=0

ode7:=x21ⅆ2ⅆx2yx2m+3xⅆⅆxyx+λyx=0

(18)

odeadvisorode7

_Gegenbauer

(19)

dsolveode7

yx=_C1x2154+12mLegendrePm2λ+4m+412,52+m,x+_C2x2154+12mLegendreQm2λ+4m+412,52+m,x

(20)

See Also

DESol, dsolve, odeadvisor, odeadvisor,TYPES


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