NoDependentVariable - Maple Help

Student[ODEs][ReduceOrder]

 NoDependentVariable
 Reduce the order of an ODE missing the dependent variable

 Calling Sequence NoDependentVariable(ODE) NoDependentVariable(ODE, y(x)) NoDependentVariable(ODE, y(x), u(t))

Parameters

 ODE - ordinary differential equation missing the dependent variable y - name; the dependent variable of the original ODE x - name; the independent variable of the original ODE u - name; the dependent variable for the reduced ODE t - name; the independent variable for the reduced ODE

Description

 • NoDependentVariable finds the solution of an ODE which doesn't contain the dependent variable other than in its derivatives.
 • The second argument, y(x), representing the variable for the existing ODE, is optional. It must be given however, if the dependent and independent variables cannot be determined from the ODE.
 • The third argument, u(t), representing the variable for the reduced ODE, is optional. If it is not given, new independent and dependent variables will be chosen which do not conflict with the existing variables.
 • The default output is a sequence consisting of the reduced ODE in terms of the new variables, followed by the transformation used to recover the original ODE from the reduced ODE.
 • If an extra option solve or solve=true is also given, an attempt is made to solve the reduced ODE and return the general solution to the original ODE. If successful, the general solution of the original ODE will be returned.

Examples

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

Given a 2nd order ODE for y(x) which doesn't depend on y(x):

 > $\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right),x,x\right)+{\mathrm{diff}\left(y\left(x\right),x\right)}^{2}x=0$
 ${\mathrm{ode}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{2}}{}{x}{=}{0}$ (1)

Find a reduction of order to a 1st order ODE, and the associated transformation:

 > $\mathrm{reduced_ode},\mathrm{tr}≔\mathrm{NoDependentVariable}\left(\mathrm{ode}\right)$
 ${\mathrm{reduced_ode}}{,}{\mathrm{tr}}{≔}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}\right){+}{{u}{}\left({t}\right)}^{{2}}{}{t}{=}{0}{,}\left\{{t}{=}{x}{,}{u}{}\left({t}\right){=}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right\}$ (2)

Solve the reduced order ODE:

 > $\mathrm{reduced_sol}≔\mathrm{Solve}\left(\mathrm{reduced_ode},u\left(t\right)\right)$
 ${\mathrm{reduced_sol}}{≔}{u}{}\left({t}\right){=}{-}\frac{{2}}{{-}{{t}}^{{2}}{+}{2}{}{\mathrm{_C1}}}$ (3)

Apply the transformation to find a simpler ODE for the original variable y(x):

 > $\mathrm{new_ode}≔\mathrm{eval}\left(\mathrm{reduced_sol},\mathrm{tr}\right)$
 ${\mathrm{new_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{-}\frac{{2}}{{-}{{x}}^{{2}}{+}{2}{}{\mathrm{_C1}}}$ (4)

Solve the simpler ODE to find the general solution of the original ODE:

 > $\mathrm{Solve}\left(\mathrm{new_ode},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{-}\frac{\sqrt{{2}}{}{\mathrm{arctanh}}{}\left(\frac{{x}{}\sqrt{{2}}}{{2}{}\sqrt{{\mathrm{_C1}}}}\right){-}{\mathrm{_C2}}{}\sqrt{{\mathrm{_C1}}}}{\sqrt{{\mathrm{_C1}}}}$ (5)

Alternatively, the original ODE can be solved in a single step if we use the option solve:

 > $\mathrm{NoDependentVariable}\left(\mathrm{ode},'\mathrm{solve}'\right)$
 ${y}{}\left({x}\right){=}{-}\frac{\sqrt{{2}}{}{\mathrm{arctanh}}{}\left(\frac{{x}{}\sqrt{{2}}}{{2}{}\sqrt{{\mathrm{_C1}}}}\right)}{\sqrt{{\mathrm{_C1}}}}{+}{\mathrm{_C2}}$ (6)

Another example:

 > $\mathrm{ode2}≔\mathrm{diff}\left(y\left(x\right),x,x\right)+x\mathrm{diff}\left(y\left(x\right),x\right)=0$
 ${\mathrm{ode2}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){=}{0}$ (7)
 > $\mathrm{reduced_ode2},\mathrm{tr2}≔\mathrm{NoDependentVariable}\left(\mathrm{ode2}\right)$
 ${\mathrm{reduced_ode2}}{,}{\mathrm{tr2}}{≔}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}\right){+}{t}{}{u}{}\left({t}\right){=}{0}{,}\left\{{t}{=}{x}{,}{u}{}\left({t}\right){=}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right\}$ (8)
 > $\mathrm{reduced_sol2}≔\mathrm{Solve}\left(\mathrm{reduced_ode2},u\left(t\right)\right)$
 ${\mathrm{reduced_sol2}}{≔}{u}{}\left({t}\right){=}{{ⅇ}}^{{-}\frac{{{t}}^{{2}}}{{2}}{+}{\mathrm{_C1}}}$ (9)
 > $\mathrm{new_ode2}≔\mathrm{eval}\left(\mathrm{reduced_sol2},\mathrm{tr2}\right)$
 ${\mathrm{new_ode2}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{{ⅇ}}^{{-}\frac{{{x}}^{{2}}}{{2}}{+}{\mathrm{_C1}}}$ (10)
 > $\mathrm{Solve}\left(\mathrm{new_ode2},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}\frac{{{ⅇ}}^{{\mathrm{_C1}}}{}\sqrt{{\mathrm{\pi }}}{}\sqrt{{2}}{}{\mathrm{erf}}{}\left(\frac{\sqrt{{2}}{}{x}}{{2}}\right)}{{2}}{+}{\mathrm{_C2}}$ (11)
 > $\mathrm{NoDependentVariable}\left(\mathrm{ode2},'\mathrm{solve}'\right)$
 ${y}{}\left({x}\right){=}\frac{{{ⅇ}}^{{\mathrm{_C1}}}{}\sqrt{{\mathrm{\pi }}}{}\sqrt{{2}}{}{\mathrm{erf}}{}\left(\frac{\sqrt{{2}}{}{x}}{{2}}\right)}{{2}}{+}{\mathrm{_C2}}$ (12)

A third example:

 > $\mathrm{ode3}≔\mathrm{diff}\left(y\left(x\right),x,x\right)=\mathrm{exp}\left(x\right)\mathrm{diff}\left(y\left(x\right),x\right)$
 ${\mathrm{ode3}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{{ⅇ}}^{{x}}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (13)
 > $\mathrm{reduced_ode3},\mathrm{tr3}≔\mathrm{NoDependentVariable}\left(\mathrm{ode3}\right)$
 ${\mathrm{reduced_ode3}}{,}{\mathrm{tr3}}{≔}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}\right){=}{{ⅇ}}^{{t}}{}{u}{}\left({t}\right){,}\left\{{t}{=}{x}{,}{u}{}\left({t}\right){=}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right\}$ (14)
 > $\mathrm{reduced_sol3}≔\mathrm{Solve}\left(\mathrm{reduced_ode3},u\left(t\right)\right)$
 ${\mathrm{reduced_sol3}}{≔}{u}{}\left({t}\right){=}{{ⅇ}}^{{{ⅇ}}^{{t}}{+}{\mathrm{_C1}}}$ (15)
 > $\mathrm{new_ode3}≔\mathrm{eval}\left(\mathrm{reduced_sol3},\mathrm{tr3}\right)$
 ${\mathrm{new_ode3}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{{ⅇ}}^{{{ⅇ}}^{{x}}{+}{\mathrm{_C1}}}$ (16)
 > $\mathrm{Solve}\left(\mathrm{new_ode3},y\left(x\right)\right)$
 ${y}{}\left({x}\right){=}{-}{{ⅇ}}^{{\mathrm{_C1}}}{}{{\mathrm{Ei}}}_{{1}}{}\left({-}{{ⅇ}}^{{x}}\right){+}{\mathrm{_C2}}$ (17)
 > $\mathrm{NoDependentVariable}\left(\mathrm{ode3},'\mathrm{solve}'\right)$
 ${y}{}\left({x}\right){=}{-}{{ⅇ}}^{{\mathrm{_C1}}}{}{{\mathrm{Ei}}}_{{1}}{}\left({-}{{ⅇ}}^{{x}}\right){+}{\mathrm{_C2}}$ (18)

Compatibility

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