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:

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\left[\mathrm{Solve}\right]\right)\:$

$\mathrm{ode1}≔x^3\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)+3x^2\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-6x\mathrm{diff}\left(y\left(x\right)\,x\right)-6y\left(x\right)=0$

$\mathrm{ode1}≔x^3\left(\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)\right)+3x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-6x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)-6y\left(x\right)=0$

$\mathrm{CauchyEuler}\left(\mathrm{ode1}\,y\left(x\right)\right)$

$y\left(x\right)=3\mathrm{\_C1}{x}^3-\frac{\mathrm{\_C2}}{x}-\frac{2\mathrm{\_C3}}{x^2}$

$\mathrm{ode2}≔x^2\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-4x\mathrm{diff}\left(y\left(x\right)\,x\right)+2y\left(x\right)=0$

$\mathrm{ode2}≔x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-4x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+2y\left(x\right)=0$

$\mathrm{CauchyEuler}\left(\mathrm{ode2}\,y\left(x\right)\right)$

$y\left(x\right)=\mathrm{\_C1}{\ⅇ}^{-\frac{\left(-5+\sqrt{17}\right)\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%ln}\left(x\right)\right]\right)}{2}}+\mathrm{\_C2}{\ⅇ}^{\frac{\left(5+\sqrt{17}\right)\mathrm{Typesetting}:-\mathrm{\_Hold}\left(\left[\mathrm{\%ln}\left(x\right)\right]\right)}{2}}$

$\mathrm{ode3}≔-x^2\mathrm{diff}\left(y\left(x\right)\,x\,x\right)-x\mathrm{diff}\left(y\left(x\right)\,x\right)+9y\left(x\right)=0$

$\mathrm{ode3}≔-x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)-x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+9y\left(x\right)=0$

$\mathrm{CauchyEuler}\left(\mathrm{ode3}\,y\left(x\right)\right)$

$y\left(x\right)=\frac{\mathrm{\_C1}}{x^3}+\mathrm{\_C2}{x}^3$

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

For more information on Maple 2021 changes, see Updates in Maple 2021.