integrate the solutions of a differential operator or equation

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DEtools[integrate_sols] - integrate the solutions of a differential operator or equation

	Calling Sequence
	integrate_sols(L, domain) integrate_sols(eqn, dvar)

Parameters

L	-	differential operator
domain	-	list containing two names
eqn	-	homogeneous linear differential equation
dvar	-	dependent variable

Description

•	The input L is a differential operator. This procedure computes an operator M of minimal order such that any solution of L has an antiderivative which is a solution of M.

•	If the order of L equals the order of M then the output is a list [M, r] such that r(f) is an antiderivative of and also a solution of M for every solution of L. If the order of L is not equal to M then only M is given in the output. In this case M equals where is the derivation.

•	The argument domain describes the differential algebra. If this argument is the list , then the differential operators are notated with the symbols and . They are viewed as elements of the differential algebra where is the field of constants.

•	If the argument domain is omitted then the differential specified by the environment variable _Envdiffopdomain is used. If this environment variable is not set then the argument domain may not be omitted.

•	Instead of a differential operator, the input can also be a linear homogeneous differential equation having rational function coefficients. In this case the second argument must be the dependent variable.

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

Examples

(1)

(2)

The result was only the operator M. For an example where both M and r are obtained, consider:

L := Dz^2-3*Dz/(2*z*(z-1))-(z-2)*(z^2-z+1)/(z*(z-1)^2)

L := Dz^2-(3/2)*Dz/(z*(z-1))-(z-2)*(z^2-z+1)/(z*(z-1)^2)

(3)

[-z*Dz^2/(z-1)+(1/2)*Dz/(z-1)^2+1, z*Dz/(z-1)-(1/2)/(z-1)^2]

(4)

Regarding the meaning of the second element in the output of integrate_sols, consider the following second order ODE and its solution.

ode := (27*x^2+4)*(diff(y(x), `$`(x, 2)))+27*x*(diff(y(x), x))-3*y(x)

(5)

sol := y(x) = _C1*sin((1/3)*arctan(3*3^(1/2)*x/(-27*x^2-4)^(1/2)))+_C2*cos((1/3)*arctan(3*3^(1/2)*x/(-27*x^2-4)^(1/2)))

(6)

The ODE satisfied by the integral of sol is given by the first element of the output of integrate_sols while the formula for computing its solution (as a function of sol) is given by the second element.

ode2 := y(x)-(9/8)*x*(diff(y(x), x))+((9/8)*x^2+1/6)*(diff(y(x), `$`(x, 2)))

(7)

(8)

So the solution to ode2 is obtained by substituting sol into integral_of_sol.

(9)

(10)

That sol2 is the integral of sol can also be verified as follows: Differentiate sol2 and verify that it is equal to sol.

(11)

	See Also
	DESols, diffop

References

Abramov, S.A., and van Hoeij, M. "A method for the Integration of Solutions of Ore Equations." ISSAC '97 Proceedings, pp. 172-175. 1997.

van der Put, M., and Singer, M. F. Galois Theory of Linear Differential Equations, Vol. 328. Springer, 2003. An electronic version of this book is available at http://www4.ncsu.edu/~singer/ms_papers.html.