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 indicialeq
 compute the indicial polynomial of a homogeneous linear ODE

 Calling Sequence indicialeq(des, ivar, alpha, dvar)

Parameters

 des - homogeneous linear ordinary differential equation or its list form ivar - independent variable alpha - point dvar - (optional) dependent variable

Description

 • This routine is used to determine either the indicial equation at the point $\mathrm{ivar}=\mathrm{\alpha }$ for a given homogeneous linear ordinary differential equation, or an equivalent list form (either as generated by DEtools[convertAlg] or as a coefficient list). If the ODE is regular singular (also called Fuchsian) at $\mathrm{ivar}=\mathrm{\alpha }$, then the degree of the indicial equation equals the order of the ODE; otherwise it is smaller than the order of the ODE. The roots of the indicial equation are called the exponents of the ODE at $\mathrm{ivar}=\mathrm{\alpha }$.
 • des may be in the standard differential equation form, or in one of two list forms: 1) a list as generated by DEtools[convertAlg], or 2) a coefficient list (as in the first element of a list generated by DEtools[convertAlg]).
 • dvar must be specified when des is input as a differential equation.
 • The definition of indicial equation is as follows. Let $n$ be a new variable, substitute $\mathrm{dvar}={\left(\mathrm{ivar}-\mathrm{\alpha }\right)}^{n}$ into the equation, take the series expansion at $\mathrm{ivar}=\mathrm{\alpha }$, and take the coefficient of the first non-zero term. The result will be a polynomial in $n$, which by definition is the indicial equation (indicialeq replaces $n$ by ivar so that no new variable is needed, and also divides by the leading coefficient). This definition is illustrated by an example below.
 • This function is part of the DEtools package, and so it can be used in the form indicialeq(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[indicialeq](..).

Examples

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

Coefficient list form:

 > $\mathrm{coefs}≔\left[21\left({x}^{2}-x+1\right),0,100{x}^{2}{\left(x-1\right)}^{2}\right]:$
 > $\mathrm{indicialeq}\left(\mathrm{coefs},x,0\right)$
 ${{x}}^{{2}}{-}{x}{+}\frac{{21}}{{100}}{=}{0}$ (1)
 > $\mathrm{indicialeq}\left(\mathrm{coefs},x,1\right)$
 ${{x}}^{{2}}{-}{x}{+}\frac{{21}}{{100}}{=}{0}$ (2)

Differential equation form:

 > $\mathrm{DE}≔-\left(\frac{72\cdot 11}{73}\left(x-1\right)+\frac{5}{8}+\frac{2}{63}\right)y\left(x\right)+\left(\frac{72}{7}x\left(x-1\right)-\frac{20}{9}\left(x-1\right)+\frac{3x}{4}\right)\mathrm{diff}\left(y\left(x\right),x\right)+x\left(7x-4\right)\left(x-1\right)\mathrm{diff}\left(y\left(x\right),\mathrm{}\left(x,2\right)\right):$
 > $\mathrm{indicialeq}\left(\mathrm{DE},x,\frac{4}{7},y\left(x\right)\right)$
 ${{x}}^{{2}}{-}\frac{{593}}{{1764}}{}{x}{=}{0}$ (3)
 > $\mathrm{indicialeq}\left(\mathrm{DE},x,1,y\left(x\right)\right)$
 ${{x}}^{{2}}{-}\frac{{3}}{{4}}{}{x}{=}{0}$ (4)
 > $\mathrm{indicialeq}\left(\mathrm{DE},x,0,y\left(x\right)\right)$
 ${{x}}^{{2}}{-}\frac{{4}}{{9}}{}{x}{=}{0}$ (5)

This could also have been computed directly from the definition of the indicial equation, as follows:

 > $P≔\mathrm{eval}\left(\mathrm{DE},y\left(x\right)={x}^{n}\right):$
 > $P≔\mathrm{series}\left(\mathrm{normal}\left(\frac{P}{{x}^{n}}\right),x=0\right):$
 > $P≔\mathrm{subs}\left(n=x,\mathrm{tcoeff}\left(\mathrm{convert}\left(P,\mathrm{polynom}\right),x\right)\right):$
 > $\frac{P}{\mathrm{lcoeff}\left(P,x\right)}$
 ${{x}}^{{2}}{-}\frac{{4}}{{9}}{}{x}$ (6)

ConvertAlg form:

 > $Y≔\left(2{x}^{2}+5{x}^{3}\right)\mathrm{diff}\left(y\left(x\right),x,x\right)+\left(5x-{x}^{2}\right)\mathrm{diff}\left(y\left(x\right),x\right)+\left(1+x\right)y\left(x\right)=0:$
 > $Y≔\mathrm{convertAlg}\left(Y,y\left(x\right)\right)$
 ${Y}{≔}\left[\left[{1}{+}{x}{,}{-}{{x}}^{{2}}{+}{5}{}{x}{,}{5}{}{{x}}^{{3}}{+}{2}{}{{x}}^{{2}}\right]{,}{0}\right]$ (7)
 > $\mathrm{indicialeq}\left(Y,x,-\frac{2}{5},y\left(x\right)\right)$
 ${{x}}^{{2}}{-}\frac{{37}}{{10}}{}{x}{=}{0}$ (8)
 > $\mathrm{indicialeq}\left(Y,x,0,y\left(x\right)\right)$
 ${{x}}^{{2}}{+}\frac{{3}}{{2}}{}{x}{+}\frac{{1}}{{2}}{=}{0}$ (9)

References

  Ince, E.L. Ordinary Differential Equations, Chapters XVI and XVII. New York: Dover Publications, 1956.