Linear Differential Operators
Description
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
${a}_{0}y\left(x\right)\+{a}_{1}\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\+...\+{a}_{n}\left(\frac{{\ⅆ}^{n}}{{\ⅆx}^{n}}y\left(x\right)\right)\=0.$
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.
See Also
DEtools
DEtools[de2diffop]
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