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:

Use the option output=steps to make this command return an annotated step-by-step solution.  Further control over the format and display of the step-by-step solution is available using the options described in Student:-Basics:-OutputStepsRecord.  The options supported by that command can be passed to this one.

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

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

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

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

$y\left(x\right)=\frac{\mathrm{c\_\_1}{x}^5+\mathrm{c\_\_3}x+\mathrm{c\_\_2}}{x^2}$

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

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

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

$y\left(x\right)=x^{\frac{5}{2}-\frac{\sqrt{17}}{2}}\left(x^{\sqrt{17}}\mathrm{c\_\_2}+\mathrm{c\_\_1}\right)$

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

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

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

$y\left(x\right)=\frac{\mathrm{c\_\_1}}{x^3}+\mathrm{c\_\_2}{x}^3$

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

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

The Student[ODEs][Solve][CauchyEuler] command was updated in Maple 2022.

The output option was introduced in Maple 2022.

For more information on Maple 2022 changes, see Updates in Maple 2022.