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DEtools

  

firtest

  

test a given first integral

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

firtest(first_int, ODE, y(x))

Parameters

first_int

-

first integral

ODE

-

ordinary differential equation

y(x)

-

(optional) indeterminate function of the ODE

Description

• 

The firtest command checks whether a given expression is a first integral of a given ODE. Similar to odetest, firtest returns 0 when the result is valid, or an algebraic expression obtained after simplifying the PDE for the first integral associated with the given ODE (see odepde). Among other things, firtest can be used to test the results obtained using the command firint.

• 

If the result returned by firtest is not zero, the expression might nevertheless be a first integral. Sometimes, with further simplification, you can obtain the desired 0 using commands such as expand, combine, and so on.

• 

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

Examples

A first order ODE

withDEtools:

ODEⅆⅆxyx=yxax

ODE:=ⅆⅆxyx=yxax

(1)

An integrating factor for ODE above

ΜintfactorODE

Μ:=ⅇ∫axⅆx

(2)

A related (to Mu) first integral for ODE above

ansfirintΜODE

ans:=ⅇ∫axⅆxyx+_C1=0

(3)

Testing the first integral

firtestans,ODE

0

(4)

A second order ODE example

ODEⅆ2ⅆx2yx=22ⅆⅆxyx+5xyx2+2x2yxⅆⅆxyxx

ODE:=ⅆ2ⅆx2yx=22ⅆⅆxyx+5xyx2+2x2yxⅆⅆxyxx

(5)

first_int2x5yx2+x4ⅆⅆxyx+_C1=0

first_int:=2x5yx2+x4ⅆⅆxyx+_C1=0

(6)

firtestfirst_int,ODE

0

(7)

See Also

DEtools

dsolve

firint

intfactor

PDEtools

redode

 


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