ChangeVariables(tr, ODE, y(x), u(t)) applies the change of variables tr to the given ODE.

The second and third arguments, y(x) and u(t), representing the existing and new dependent variables, are optional; however they must be given if the existing and dependent variables cannot be determined from the form of the transformation.

There must be two transformation equations, each of which specifies the new form of one of the following three entities:

If the left-hand sides of the transformation equations are x and diff(y(x),x), then the ODE should not contain y(x) (independently of diff(y(x),x)). If the left-hand sides of the transformation equations are y(x) and diff(y(x),x), then the ODE should not contain x (independently of y(x) and diff(y(x),x)).

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

$\mathrm{ode1}≔\frac{x^2\left(y\left(x\right)+1\right)}{y\left(x\right)}+\left(x-1\right)\mathrm{diff}\left(y\left(x\right)\,x\right)=x^2\left(y\left(x\right)+\frac{1}{y\left(x\right)}\right)$

$\mathrm{ode1}≔\frac{x^2\left(y\left(x\right)+1\right)}{y\left(x\right)}+\left(x-1\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)=x^2\left(y\left(x\right)+\frac{1}{y\left(x\right)}\right)$

$\mathrm{ChangeVariables}\left(\left\{x=u\left(t\right)\,y\left(x\right)=t\right\}\,\mathrm{ode1}\right)$

$\frac{{u\left(t\right)}^2\left(t+1\right)}{t}+\frac{u\left(t\right)-1}{\frac{\ⅆ}{\ⅆt}u\left(t\right)}={u\left(t\right)}^2\left(t+\frac{1}{t}\right)$

$\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=\frac{x^2}{x-1}\mathrm{diff}\left(y\left(x\right)\,x\right)-\frac{x^2}{x-1}$

$\mathrm{ode2}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\frac{x^2\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{x-1}-\frac{x^2}{x-1}$

$\mathrm{ChangeVariables}\left(\left\{x=t\,\mathrm{diff}\left(y\left(x\right)\,x\right)=u\left(t\right)\right\}\,\mathrm{ode2}\right)$

$\frac{\ⅆ}{\ⅆt}u\left(t\right)=\frac{t^{2}u\left(t\right)}{t-1}-\frac{t^2}{t-1}$

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

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