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particularsol

  

find a particular solution to a nonlinear ODE, or a linear non-homogeneous ODE, without computing its general solution

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

particularsol(ODE, y(x))

particularsol(ODE_coeff_list, x)

Parameters

ODE

-

non-linear, or linear non-homogeneous ODE

y(x)

-

dependent variable (required only when not obvious); can be any unknown function of one variable

ODE_coeff_list

-

list of coefficients representing a non-homogeneous linear ODE (see DEtools[convertAlg])

x

-

independent variable

Description

• 

The particularsol routine is used to find a particular solution for a nonlinear ordinary differential equation (ODE), or for a non-homogeneous linear ODE without computing the general solution to its homogeneous part.

• 

For nonlinear ODEs, the approach used is to compute symmetries and from there compute invariant (particular) solutions.

• 

For linear ODEs, particularsol tries, in sequence, to compute particular solutions of rational (see DEtools[ratsols]), exponential and d'Alembertian form (see LinearOperators[dAlembertianSolver]). If no particular solution is found, particularsol returns NULL.

• 

When the input is an ODE, the output of particularsol is as dsolve's output, that is, an equation with the unknown yx on the left-hand-side and the (particular) solution on the right-hand-side.

• 

When the input is a list of the coefficients of yx and its derivatives representing a linear ODE, for instance obtained from the ODE using DEtools[convertAlg], the output is not an equation but an expression representing the particular solution - see the examples.

• 

In the case of a linear ODE, while particularsol is useful to compute a particular solution when the general solution of the homogeneous part of the ODE is not known, a particular solution can always be computed if that general solution is known - for that purpose use DEtools[varparam].

• 

This function is part of the DEtools package, and so it can be used in the form particularsol(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[particularsol](..).

Examples

withDEtools:

A third order non-linear ODE

odeⅆ3ⅆx3yxⅆ2ⅆx2yxyx+ⅆⅆxyx2=0

odeⅆ3ⅆx3yxⅆ2ⅆx2yxyx+ⅆⅆxyx2=0

(1)

No general solution is known for ode. A particular solution for it, however, depending on one arbitrary constant, is computable via

particularsolode

yx=0,yx=6x+_C1

(2)

A second order non-homogeneous linear ODE whose homogeneous part is out of the scope of dsolve and a particular rational solution for it

NH_ODE133601600831056948930009536x3+920184312063950707048658496x2+4242610768571010035328624653+3711155578640470803611376x4369364944765014363775910745x34028299868602335386591950537x27123173270557941102013850561xyx+158668797683584879758300854361937570512208809606272361x4743569265631618257116222299x3+253413657274110944226342787x2+698563894241740551690579371xⅆⅆxyx+31950844168887177328078428089797769268184221773945x+14158747037070037877844107x2+8089797769268184221773945x310963662620181320145036265x4ⅆ2ⅆx2yx

NH_ODE133601600831056948930009536x3+920184312063950707048658496x2+3711155578640470803611376x4369364944765014363775910745x34028299868602335386591950537x27123173270557941102013850561x4242610768571010035328624653yx+361937570512208809606272361x4743569265631618257116222299x3+253413657274110944226342787x2+698563894241740551690579371x+158668797683584879758300854ⅆⅆxyx+10963662620181320145036265x4+8089797769268184221773945x3+14158747037070037877844107x28089797769268184221773945x3195084416888717732807842ⅆ2ⅆx2yx

(3)

p_solparticularsolNH_ODE

p_solyx=4x49x4+35x3+49x2+21x+6x212x15

(4)

odetestp_sol,NH_ODE

0

(5)

Represent this non-homogeneous linear ODE using a list with the coefficients of y(x) and its derivatives

coeff_listDEtoolsconvertAlgNH_ODE,yx

coeff_list3711155578640470803611376x4369364944765014363775910745x34028299868602335386591950537x27123173270557941102013850561x4242610768571010035328624653,361937570512208809606272361x4743569265631618257116222299x3+253413657274110944226342787x2+698563894241740551690579371x+158668797683584879758300854,10963662620181320145036265x4+8089797769268184221773945x3+14158747037070037877844107x28089797769268184221773945x3195084416888717732807842,133601600831056948930009536x3920184312063950707048658496x2

(6)

p_solparticularsolcoeff_list,x

p_sol4x49x4+35x3+49x2+21x+6x212x15

(7)

Note particularsol is not restricted to the computation of rational particular solutions

odeⅆ2ⅆx2yxyxx=4x3+1lnxxx

odeⅆ2ⅆx2yxyxx=4x3+1lnxx32

(8)

p_solparticularsolode

p_solyx=4xlnx

(9)

odetestp_sol,ode

0

(10)

See Also

DEtools

DEtools[convertAlg]

DEtools[varparam]

dsolve

odetest