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 $f$ and also a solution of M for every solution $f$ 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 $L\mathrm{Dt}$ where $\mathrm{Dt}$ is the derivation.

The argument domain describes the differential algebra. If this argument is the list $\left[\mathrm{Dt}\,t\right]$, then the differential operators are notated with the symbols $\mathrm{Dt}$ and $t$. They are viewed as elements of the differential algebra $C\left(t\right)$ $\left[\mathrm{Dt}\right]$ where $C$ 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](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$L≔\mathrm{Dz}+\frac{1}{z}$

$L≔\mathrm{Dz}+\frac{1}{z}$

$\mathrm{integrate\_sols}\left(L\,\left[\mathrm{Dz}\,z\right]\right)$

$\left(\mathrm{Dz}+\frac{1}{z}\right)\mathrm{Dz}$

$L≔{\mathrm{Dz}}^2-\frac{3}{2z\left(z-1\right)}\mathrm{Dz}-\frac{\left(z-2\right)\left(z^2-z+1\right)}{z{\left(z-1\right)}^2}$

$L≔{\mathrm{Dz}}^2-\frac{3\mathrm{Dz}}{2z\left(z-1\right)}-\frac{\left(z-2\right)\left(z^2-z+1\right)}{z{\left(z-1\right)}^2}$

$\mathrm{integrate\_sols}\left(L\,\left[\mathrm{Dz}\,z\right]\right)$

$\left[-\frac{z{\mathrm{Dz}}^2}{z-1}+\frac{\mathrm{Dz}}{2{\left(z-1\right)}^2}+1\,\frac{z\mathrm{Dz}}{z-1}-\frac{1}{2{\left(z-1\right)}^2}\right]$

$\mathrm{ode}≔\left(27x^2+4\right)\mathrm{diff}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)\,x\right)+27x\mathrm{diff}\left(y\left(x\right)\,x\right)-3y\left(x\right)$

$\mathrm{ode}≔\left(27x^2+4\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+27x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-3y\left(x\right)$

$\mathrm{sol}≔\mathrm{dsolve}\left(\mathrm{ode}\,y\left(x\right)\right)$

$\mathrm{sol}≔y\left(x\right)=\mathrm{c\_\_1}\sinh \left(\frac{\mathrm{arcsinh}\left(\frac{3\sqrt{3}x}{2}\right)}{3}\right)+\mathrm{c\_\_2}\cosh \left(\frac{\mathrm{arcsinh}\left(\frac{3\sqrt{3}x}{2}\right)}{3}\right)$

$\mathrm{int\_s}≔\mathrm{integrate\_sols}\left(\mathrm{ode}\,y\left(x\right)\right)\:$

$\mathrm{ode2}≔\mathrm{int\_s}\left[1\right]$

$\mathrm{ode2}≔y\left(x\right)-\frac{9x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{8}+\left(\frac{9x^2}{8}+\frac{1}{6}\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

$\mathrm{integral\_of\_sol}≔\mathrm{int\_s}\left[2\right]$

$\mathrm{integral\_of\_sol}≔\frac{9xy\left(x\right)}{8}+\left(-\frac{9x^2}{8}-\frac{1}{6}\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)$

$\mathrm{sol2}≔y\left(x\right)=\mathrm{simplify}\left(\mathrm{eval}\left(\mathrm{integral\_of\_sol}\,\mathrm{sol}\right)\right)$

$\mathrm{sol2}≔y\left(x\right)=\frac{9x\left(\mathrm{c\_\_1}\sinh \left(\frac{\mathrm{arcsinh}\left(\frac{3\sqrt{3}x}{2}\right)}{3}\right)+\mathrm{c\_\_2}\cosh \left(\frac{\mathrm{arcsinh}\left(\frac{3\sqrt{3}x}{2}\right)}{3}\right)\right)}{8}-\frac{\sqrt{27x^2+4}\sqrt{3}\left(\mathrm{c\_\_1}\cosh \left(\frac{\mathrm{arcsinh}\left(\frac{3\sqrt{3}x}{2}\right)}{3}\right)+\mathrm{c\_\_2}\sinh \left(\frac{\mathrm{arcsinh}\left(\frac{3\sqrt{3}x}{2}\right)}{3}\right)\right)}{24}$

$\mathrm{odetest}\left(\mathrm{sol2}\,\mathrm{ode2}\right)$

$0$

$\mathrm{simplify}\left(\mathrm{diff}\left(\mathrm{rhs}\left(\mathrm{sol2}\right)\,x\right)-\mathrm{rhs}\left(\mathrm{sol}\right)\right)$

$0$

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.