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dsolve - find series solutions to ODE problems

Calling Sequence

dsolve(ODE, y(x), 'series')

dsolve(ODE, y(x), 'series', x=pt)

dsolve({ODE, ICs}, y(x), 'series')

dsolve({sysODE, ICs}, {funcs}, 'series')

dsolve(ODE, y(x), 'type=series')

dsolve(ODE, y(x), 'type=series', x=pt)

dsolve({ODE, ICs}, y(x), 'type=series')

dsolve({sysODE, ICs}, {funcs}, 'type=series')

Parameters

ODE

-

ordinary differential equation

y(x)

-

dependent variable (indeterminate function)

ICs

-

initial conditions for y(x) and/or its derivatives

sysODE

-

system of ODEs

{funcs}

-

set with indeterminate functions

pt

-

expansion point for series

'type=series'

-

to request a series solution

Description

• 

The dsolve command uses several methods when trying to find a series solution to an ODE or a system of ODEs. When initial conditions or an expansion point are given, the series is calculated at the given point; otherwise, the series is calculated at the origin.

• 

The first method used is a Newton iteration based on a paper of Keith Geddes. See the References section in this help page.

• 

The second method involves a direct substitution to generate a system of equations, which may be solvable (by solve) to give a series.

• 

The third method is the method of Frobenius for nth order linear DEs. See the References section in this help page.

• 

If the aforementioned methods fail, the function invokes LinearFunctionalSystems[SeriesSolution].

Examples

ode:=ⅆ2ⅆt2yt+ⅆⅆtyt2=0

ode:=ⅆ2ⅆt2yt+ⅆⅆtyt2=0

(1)

When the initial conditions are not given, the answer is expressed in terms of the indeterminate function and its derivatives evaluated at the origin.

ans:=dsolveode,yt,type='series'

ans:=yt=y0+Dy0t12Dy02t2+13Dy03t314Dy04t4+15Dy05t5+Ot6

(2)

If initial conditions are given, the series is calculated at that the given point:

ans:=dsolveode,ya=Y_a,Dya=DY_a,yt,type='series'

ans:=yt=Y_a+DY_ata12DY_a2ta2+13DY_a3ta314DY_a4ta4+15DY_a5ta5+Ota6

(3)

Alternatively, an expansion point can be provided, which is most useful when initial conditions cannot be given:

ans:=dsolve1t2ⅆ2ⅆt2yt2tytyt,yt,'series',t=1

ans:=yt=_C1t1134t1+748t12+1128t1315715360t14+3371921600t15+Ot16+_C2lnt132t1+98t12732t133256t14+15710240t15+Ot16+12916t12+2132t131313072t142219102400t15+Ot16

(4)

The order of the series expansion (default = 6) can be changed using (an environment variable - see Order). For example,

Order:=3

Order:=3

(5)

An example with a system of ODEs.

sys:=ⅆⅆtyt=xt,ⅆⅆtxt=yt

sys:=ⅆⅆtxt=yt,ⅆⅆtyt=xt

(6)

ans:=dsolvesys∪x0=A,y0=B,xt,yt,type='series'

ans:=xt=A+Bt12At2+Ot3,yt=BAt12Bt2+Ot3

(7)

An example solved by LinearFunctionalSystems[SeriesSolution].

sys:=ⅆⅆxy1xy1x+xy2x=x3,xⅆⅆxy2x2y2x

sys:=ⅆⅆxy1xy1x+xy2x=x3,xⅆⅆxy2x2y2x

(8)

vars:=y1x,y2x

vars:=y1x,y2x

(9)

dsolveopsys∪y10=13,vars,'series'

y1x=13+13x+132x2+Ox3,y2x=12D2y20x2+Ox3

(10)

See Also

DEtools, dsolve, dsolve,ICs, dsolve,inttrans, dsolve,Lie, dsolve,numeric, dsolve,system, LinearFunctionalSystems[SeriesSolution], plots[odeplot], series, Slode

References

  

Forsyth, A.R. Theory of Differential Equations. Cambridge: University Press, 1906. pp. 78-90

  

Geddes, Keith.  "Convergence Behaviour of the Newton Iteration for First Order Differential Equations". Proceedings of EUROSAM '79. pp.189-199.

  

Ince, E.L. Ordinary Differential Equations. Dover Publications, 1956. pp. 398-406.


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