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 clairautsol
 find solutions of a Clairaut first order ODE

 Calling Sequence clairautsol(lode, v)

Parameters

 lode - first order differential equation v - dependent variable of the lode

Description

 • The clairautsol routine determines if the first argument is a first order ODE of Clairaut type and, if so, returns a solution to the equation.
 • The first argument is a differential equation in diff or D form and the second argument is the function in the differential equation.
 • This function is part of the DEtools package, and so it can be used in the form clairautsol(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[clairautsol](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $\mathrm{ode}≔z\left(t\right)=t\left(\frac{ⅆ}{ⅆt}z\left(t\right)\right)+{\left(\frac{ⅆ}{ⅆt}z\left(t\right)\right)}^{3}$
 ${\mathrm{ode}}{≔}{z}{}\left({t}\right){=}{t}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({t}\right)\right){+}{\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{z}{}\left({t}\right)\right)}^{{3}}$ (1)
 > $\mathrm{clairautsol}\left(\mathrm{ode},z\left(t\right)\right)$
 $\left\{{z}{}\left({t}\right){=}{-}\frac{{2}{}\sqrt{{-}{3}{}{t}}{}{t}}{{9}}{,}{z}{}\left({t}\right){=}\frac{{2}{}\sqrt{{-}{3}{}{t}}{}{t}}{{9}}{,}{z}{}\left({t}\right){=}{{\mathrm{_C1}}}^{{3}}{+}{t}{}{\mathrm{_C1}}\right\}$ (2)
 > $\mathrm{ode}≔z\left(t\right)=t\mathrm{D}\left(z\right)\left(t\right)+{\mathrm{D}\left(z\right)\left(t\right)}^{2}-{\mathrm{D}\left(z\right)\left(t\right)}^{3}$
 ${\mathrm{ode}}{≔}{z}{}\left({t}\right){=}{t}{}{\mathrm{D}}{}\left({z}\right){}\left({t}\right){+}{{\mathrm{D}}{}\left({z}\right){}\left({t}\right)}^{{2}}{-}{{\mathrm{D}}{}\left({z}\right){}\left({t}\right)}^{{3}}$ (3)
 > $\mathrm{clairautsol}\left(\mathrm{ode},z\left(t\right)\right)$
 $\left\{{z}{}\left({t}\right){=}\frac{{t}}{{3}}{+}\frac{{2}}{{27}}{-}\frac{{2}{}\sqrt{{27}{}{{t}}^{{3}}{+}{27}{}{{t}}^{{2}}{+}{9}{}{t}{+}{1}}}{{27}}{,}{z}{}\left({t}\right){=}\frac{{t}}{{3}}{+}\frac{{2}}{{27}}{+}\frac{{2}{}\sqrt{{27}{}{{t}}^{{3}}{+}{27}{}{{t}}^{{2}}{+}{9}{}{t}{+}{1}}}{{27}}{,}{z}{}\left({t}\right){=}{-}{{\mathrm{_C1}}}^{{3}}{+}{{\mathrm{_C1}}}^{{2}}{+}{t}{}{\mathrm{_C1}}\right\}$ (4)