Lagerstrom ODEs - Maple Programming Help

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Lagerstrom ODEs

Description

 • The general form of the Lagerstrom ODE is given by the following:
 > Lagerstrom_ode := diff(y(x),x,x)= -k*diff(y(x),x)/x-epsilon*y(x)*diff(y(x),x);
 ${\mathrm{Lagerstrom_ode}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{y}{}\left({x}\right){=}{-}\frac{{k}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right)}{{x}}{-}{\mathrm{ε}}{}{y}{}\left({x}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right)$ (1)
 See Rosenblat and Shepherd, "On the Asymptotic Solution of the Lagerstrom Model Equation".

Examples

The second order Lagerstrom ODE can be reduced to a first order ODE of Abel type once the system succeeds in finding one polynomial symmetry for it (see symgen):

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor},\mathrm{symgen}\right):$
 > $\mathrm{odeadvisor}\left(\mathrm{Lagerstrom_ode}\right)$
 $\left[{\mathrm{_Lagerstrom}}{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_with_linear_symmetries}}\right]\right]$ (2)
 > $\mathrm{symgen}\left(\mathrm{Lagerstrom_ode},\mathrm{way}=3\right)$
 $\left[{\mathrm{_ξ}}{=}{-}{x}{,}{\mathrm{_η}}{=}{y}\right]$ (3)

From which, giving the same indication directly to dsolve, you obtain the reduction of order

 > $\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{Lagerstrom_ode},\mathrm{way}=3\right)$
 ${\mathrm{ans}}{≔}{y}{}\left({x}\right){=}\left({\mathrm{_a}}{}{{ⅇ}}^{{∫}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}{\mathrm{_C1}}}\right){&where}\left[\left\{\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}{}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}\left({-}{{\mathrm{_a}}}^{{2}}{}{\mathrm{ε}}{-}{\mathrm{_a}}{}{k}{+}{2}{}{\mathrm{_a}}\right){}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{3}}{+}\left({-}{\mathrm{_a}}{}{\mathrm{ε}}{-}{k}{+}{3}\right){}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{2}}\right\}{,}\left\{{\mathrm{_a}}{=}{y}{}\left({x}\right){}{x}{,}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}{-}\frac{{1}}{{x}{}\left(\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{y}{}\left({x}\right)\right){}{x}{+}{y}{}\left({x}\right)\right)}\right\}{,}\left\{{x}{=}\frac{{1}}{{{ⅇ}}^{{∫}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}{\mathrm{_C1}}}}{,}{y}{}\left({x}\right){=}{\mathrm{_a}}{}{{ⅇ}}^{{∫}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}{\mathrm{_C1}}}\right\}\right]$ (4)

For the structure of the solution above see ODESolStruc. Reductions of order can also be tested with odetest

 > $\mathrm{odetest}\left(\mathrm{ans},\mathrm{Lagerstrom_ode}\right)$
 ${0}$ (5)

The reduced ODE is of Abel type and can be selected using the mouse, or as follows

 > $\mathrm{reduced_ode}≔\mathrm{op}\left(\left[2,2,1,1\right],\mathrm{ans}\right)$
 ${\mathrm{reduced_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}{}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}\left({-}{{\mathrm{_a}}}^{{2}}{}{\mathrm{ε}}{-}{\mathrm{_a}}{}{k}{+}{2}{}{\mathrm{_a}}\right){}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{3}}{+}\left({-}{\mathrm{_a}}{}{\mathrm{ε}}{-}{k}{+}{3}\right){}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{2}}$ (6)
 > $\mathrm{odeadvisor}\left(\mathrm{reduced_ode}\right)$
 $\left[{\mathrm{_Abel}}\right]$ (7)