Linear Differential Operators
<Text-field style="Heading 2" layout="Heading 2" bookmark="info">Description</Text-field> A differential operator L in C(x)[Dx] is an expression a_0*Dx^0+ ... +a_n*Dx^n where a_0, ... , a_n are elements of C(x). So it is a polynomial in Dx with rational functions as coefficients. In the functions for differential operators in the DEtools package, the names Dx and x (other names can be used as well) can be specified either by an entry called domain, or by setting _Envdiffopdomain to [Dx,x]. An element L in C(x)[Dx] corresponds to a linear homogeneous differential equation L( y(x) )=0. If L = a_0*Dx^0+ ... +a_n*Dx^n then this is the equation 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 Multiplication (see DEtools[mult]) in the ring C(x)[Dx] corresponds to composition of differential operators. So if L = mult(f,g) then L( y(x) ) = f(g( y(x) )). In particular mult(Dx,x) = x*Dx + 1. The following procedures allow only coefficients a_0, a_1, ... , a_n in C(x): DFactor, DFactorLCLM, eigenring, endomorphism_charpoly, formal_sol, gen_exp, integrate_sols. The following procedures allow more general coefficients: GCRD, LCLM, adjoint, de2diffop, diffop2de, exterior_power, leftdivision, mult, rightdivision, symmetric_product, symmetric_power. The main purpose of these functions is factorization of differential operators. Factorization is reducing a differential equation to another equation of lower order: If f = L*R then the solutions of R( y(x) ) = 0 are solutions of f( y(x) ) = 0 as well. In particular all exponential solutions (see DEtools[expsols]) are obtained by computing all right-hand factors of order 1.