The DFactorLCLM routine accepts a differential operator L with rational function coefficients as its first argument.

The routine returns an LCLM factorization, which is a list of operators [L1, .., Lr] such that L is a rational function times LCLM(L1, .., Lr), and the order of L is the sum of the orders of L1, .., Lr. The operators L1, .., Lr  may have a non-trivial factorization (use DFactor), but they do not have a non-trivial LCLM factorization.

The bases of solutions of L1, .., Lr form a basis of the solutions of L.

The second argument domain describes the differential algebra. If this argument is the list [Dx,x], then the differential operators will be notated with the symbols Dx and x. They are viewed as elements of the differential algebra C(x)[Dx] where C is the field of constants.

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

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

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

$\mathrm{\_Envdiffopdomain}≔\left[\mathrm{DF}\,x\right]\:$

$L≔{\mathrm{DF}}^4-2x{\mathrm{DF}}^2-2\mathrm{DF}+1+x^2$

$L≔{\mathrm{DF}}^4-2{\mathrm{DF}}^{2}x+x^2-2\mathrm{DF}+1$

$\mathrm{ode}≔\mathrm{diffop2de}\left(L\,y\left(x\right)\right)$

$\mathrm{ode}≔\left(x^2+1\right)y\left(x\right)-2\frac{\ⅆ}{\ⅆx}y\left(x\right)-2x\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+\frac{{\ⅆ}^4}{\ⅆx^4}y\left(x\right)$

$v≔\mathrm{DFactorLCLM}\left(L\right)$

$v≔\left[{\mathrm{DF}}^2-x-\mathrm{I}\,{\mathrm{DF}}^2-x+\mathrm{I}\right]$

$\mathrm{sol\_ode}≔\mathrm{map}\left(i\mapsto \mathrm{op}\left(\mathrm{dsolve}\left(\mathrm{diffop2de}\left(i\,y\left(x\right)\right)\,\mathrm{output}=\mathrm{basis}\right)\right)\,v\right)$

$\mathrm{sol\_ode}≔\left[\mathrm{AiryAi}\left(\mathrm{I}+x\right)\,\mathrm{AiryBi}\left(\mathrm{I}+x\right)\,\mathrm{AiryAi}\left(x-\mathrm{I}\right)\,\mathrm{AiryBi}\left(x-\mathrm{I}\right)\right]$