CauchyEuler - Maple Help

Student[ODEs][Solve]

 CauchyEuler
 Solve a Cauchy-Euler equation

 Calling Sequence CauchyEuler(ODE, y(x))

Parameters

 ODE - a Cauchy-Euler equation y - name; the dependent variable x - name; the independent variable

Description

 • The CauchyEuler(ODE, y(x)) command finds the solution of a Cauchy-Euler equation, which is a linear homogeneous ordinary differential equation of the form:

$\mathrm{ODE}≔{a}_{n}{x}^{n}{y}^{\left(n\right)}+{a}_{n-1}{x}^{n-1}{y}^{\left(n-1\right)}+\mathrm{...}+{a}_{0}y=0$

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\left[\mathrm{Solve}\right]\right):$
 > $\mathrm{ode1}≔{x}^{3}\mathrm{diff}\left(y\left(x\right),x,x,x\right)+3{x}^{2}\mathrm{diff}\left(y\left(x\right),x,x\right)-6x\mathrm{diff}\left(y\left(x\right),x\right)-6y\left(x\right)=0$
 ${\mathrm{ode1}}{≔}{{x}}^{{3}}{}\left(\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{3}{}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{6}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{6}{}{y}{}\left({x}\right){=}{0}$ (1)
 > $\mathrm{CauchyEuler}\left(\mathrm{ode1},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{3}{}{\mathrm{_C1}}{}{{x}}^{{3}}{-}\frac{{\mathrm{_C2}}}{{x}}{-}\frac{{2}{}{\mathrm{_C3}}}{{{x}}^{{2}}}$ (2)
 > $\mathrm{ode2}≔{x}^{2}\mathrm{diff}\left(y\left(x\right),x,x\right)-4x\mathrm{diff}\left(y\left(x\right),x\right)+2y\left(x\right)=0$
 ${\mathrm{ode2}}{≔}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{4}{}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{2}{}{y}{}\left({x}\right){=}{0}$ (3)
 > $\mathrm{CauchyEuler}\left(\mathrm{ode2},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{\mathrm{_C1}}{}{{ⅇ}}^{{-}\frac{\left({-}{5}{+}\sqrt{{17}}\right){}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%ln}}{}\left({x}\right)\right]\right)}{{2}}}{+}{\mathrm{_C2}}{}{{ⅇ}}^{\frac{\left({5}{+}\sqrt{{17}}\right){}{\mathrm{Typesetting}}{:-}{\mathrm{_Hold}}{}\left(\left[{\mathrm{%ln}}{}\left({x}\right)\right]\right)}{{2}}}$ (4)
 > $\mathrm{ode3}≔-{x}^{2}\mathrm{diff}\left(y\left(x\right),x,x\right)-x\mathrm{diff}\left(y\left(x\right),x\right)+9y\left(x\right)=0$
 ${\mathrm{ode3}}{≔}{-}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){-}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){+}{9}{}{y}{}\left({x}\right){=}{0}$ (5)
 > $\mathrm{CauchyEuler}\left(\mathrm{ode3},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}\frac{{\mathrm{_C1}}}{{{x}}^{{3}}}{+}{\mathrm{_C2}}{}{{x}}^{{3}}$ (6)

Compatibility

 • The Student[ODEs][Solve][CauchyEuler] command was introduced in Maple 2021.