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DEtools

 DEnormal
 return the normalized form of a DE

 Calling Sequence DEnormal(des, ivar, dvar)

Parameters

 des - ordinary differential equation or operator list ivar - independent variable dvar - dependent variable

Description

 • This routine returns the "normalized" form of a linear differential equation, a differential operator list, or output from DEtools[convertAlg]. "Normalized" (an equivalent form) means that the list entries or coefficients are expanded polynomials in ivar, with no common factor.
 • A differential operator list (for list A in this case) is defined by

 $L$ | $L\left(y\right)={A}_{1}y\left(x\right)+{A}_{2}y\text{'}\left(x\right)+\mathrm{...}+{A}_{n+1}{y}^{\left(n\right)}\left(x\right)$ .

 • dvar must be specified if des is a differential equation.  It is not needed if des is a list.
 • This function is part of the DEtools package, and so it can be used in the form DEnormal(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[DEnormal](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{DE}≔\frac{21\left(-2{x}^{3}\right)}{3{x}^{2}-5x+2}y\left(x\right)+\frac{42{x}^{2}}{x+1}x\left(x-1\right)\mathrm{D}\left(y\right)\left(x\right)+\frac{50{x}^{3}}{{\left(x-1\right)}^{3}}{\mathrm{D}}^{\left(2\right)}\left(y\right)\left(x\right)=x\mathrm{sin}\left(x\right)$
 ${\mathrm{DE}}{≔}{-}\frac{{42}{}{{x}}^{{3}}{}{y}{}\left({x}\right)}{{3}{}{{x}}^{{2}}{-}{5}{}{x}{+}{2}}{+}\frac{{42}{}{{x}}^{{3}}{}\left({x}{-}{1}\right){}{\mathrm{D}}{}\left({y}\right){}\left({x}\right)}{{x}{+}{1}}{+}\frac{{50}{}{{x}}^{{3}}{}{{\mathrm{D}}}^{\left({2}\right)}{}\left({y}\right){}\left({x}\right)}{{\left({x}{-}{1}\right)}^{{3}}}{=}{x}{}{\mathrm{sin}}{}\left({x}\right)$ (1)
 > $\mathrm{DE2}≔\mathrm{convertAlg}\left(\mathrm{DE},y\left(x\right)\right)$
 ${\mathrm{DE2}}{≔}\left[\left[{-}\frac{{42}{}{{x}}^{{3}}}{{3}{}{{x}}^{{2}}{-}{5}{}{x}{+}{2}}{,}\frac{{42}{}{{x}}^{{4}}}{{x}{+}{1}}{-}\frac{{42}{}{{x}}^{{3}}}{{x}{+}{1}}{,}\frac{{50}{}{{x}}^{{3}}}{{\left({x}{-}{1}\right)}^{{3}}}\right]{,}{x}{}{\mathrm{sin}}{}\left({x}\right)\right]$ (2)
 > $\mathrm{DEnormal}\left(\mathrm{DE},x,y\left(x\right)\right)$
 $\left({75}{}{{x}}^{{2}}{+}{25}{}{x}{-}{50}\right){}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}\left({63}{}{{x}}^{{5}}{-}{294}{}{{x}}^{{4}}{+}{546}{}{{x}}^{{3}}{-}{504}{}{{x}}^{{2}}{+}{231}{}{x}{-}{42}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{21}{}{y}{}\left({x}\right){}{{x}}^{{3}}{+}{21}{}{y}{}\left({x}\right){}{{x}}^{{2}}{+}{21}{}{y}{}\left({x}\right){}{x}{-}{21}{}{y}{}\left({x}\right){=}\frac{{3}{}{\mathrm{sin}}{}\left({x}\right){}{{x}}^{{3}}}{{2}}{-}{4}{}{\mathrm{sin}}{}\left({x}\right){}{{x}}^{{2}}{+}{2}{}{x}{}{\mathrm{sin}}{}\left({x}\right){+}{3}{}{\mathrm{sin}}{}\left({x}\right){-}\frac{{7}{}{\mathrm{sin}}{}\left({x}\right)}{{2}{}{x}}{+}\frac{{\mathrm{sin}}{}\left({x}\right)}{{{x}}^{{2}}}$ (3)
 > $\mathrm{DEnormal}\left(\mathrm{DE2},x\right)$
 $\left[\left[{-}{21}{}{{x}}^{{3}}{+}{21}{}{{x}}^{{2}}{+}{21}{}{x}{-}{21}{,}{63}{}{{x}}^{{5}}{-}{294}{}{{x}}^{{4}}{+}{546}{}{{x}}^{{3}}{-}{504}{}{{x}}^{{2}}{+}{231}{}{x}{-}{42}{,}{75}{}{{x}}^{{2}}{+}{25}{}{x}{-}{50}\right]{,}\frac{{3}{}{\mathrm{sin}}{}\left({x}\right){}{{x}}^{{5}}{-}{8}{}{\mathrm{sin}}{}\left({x}\right){}{{x}}^{{4}}{+}{4}{}{\mathrm{sin}}{}\left({x}\right){}{{x}}^{{3}}{+}{6}{}{\mathrm{sin}}{}\left({x}\right){}{{x}}^{{2}}{-}{7}{}{x}{}{\mathrm{sin}}{}\left({x}\right){+}{2}{}{\mathrm{sin}}{}\left({x}\right)}{{2}{}{{x}}^{{2}}}\right]$ (4)