ChangeVariables - Maple Help

Student[ODEs]

 ChangeVariables
 transform an ODE using a change of variables

 Calling Sequence ChangeVariables(tr, ODE) ChangeVariables(tr, ODE, y(x)) ChangeVariables(tr, ODE, y(x), u(t))

Parameters

 ODE - an ordinary differential equation tr - a set or list of two transformation equations y - name; the existing dependent variable x - name; the existing independent variable u - name; the new dependent variable t - name; the new independent variable

Description

 • 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:

$x,y\left(x\right),\frac{ⅆ}{ⅆx}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}y\left(x\right)$

 • 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)).

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{ODEs}\right]\right):$
 > $\mathrm{ode1}≔\frac{{x}^{2}\left(y\left(x\right)+1\right)}{y\left(x\right)}+\left(x-1\right)\mathrm{diff}\left(y\left(x\right),x\right)={x}^{2}\left(y\left(x\right)+\frac{1}{y\left(x\right)}\right)$
 ${\mathrm{ode1}}{≔}\frac{{{x}}^{{2}}{}\left({y}{}\left({x}\right){+}{1}\right)}{{y}{}\left({x}\right)}{+}\left({x}{-}{1}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){=}{{x}}^{{2}}{}\left({y}{}\left({x}\right){+}\frac{{1}}{{y}{}\left({x}\right)}\right)$ (1)
 > $\mathrm{ChangeVariables}\left(\left\{x=u\left(t\right),y\left(x\right)=t\right\},\mathrm{ode1}\right)$
 $\frac{{{u}{}\left({t}\right)}^{{2}}{}\left({t}{+}{1}\right)}{{t}}{+}\frac{{u}{}\left({t}\right){-}{1}}{\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}\right)}{=}{{u}{}\left({t}\right)}^{{2}}{}\left({t}{+}\frac{{1}}{{t}}\right)$ (2)
 > $\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right),x,x\right)=\frac{{x}^{2}}{x-1}\mathrm{diff}\left(y\left(x\right),x\right)-\frac{{x}^{2}}{x-1}$
 ${\mathrm{ode2}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{{x}}^{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{x}{-}{1}}{-}\frac{{{x}}^{{2}}}{{x}{-}{1}}$ (3)
 > $\mathrm{ChangeVariables}\left(\left\{x=t,\mathrm{diff}\left(y\left(x\right),x\right)=u\left(t\right)\right\},\mathrm{ode2}\right)$
 $\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}\right){=}\frac{{{t}}^{{2}}{}{u}{}\left({t}\right)}{{t}{-}{1}}{-}\frac{{{t}}^{{2}}}{{t}{-}{1}}$ (4)

Compatibility

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