Solving Abel's ODEs of the Second Kind, Class C - Maple Programming Help

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Solving Abel's ODEs of the Second Kind, Class C

Description

 • The general form of Abel's equation, second kind, class C is given by:
 > Abel_ode2C := (g1(x)*y(x)+g0(x))*diff(y(x),x) = f3(x)*y(x)^3 + f2(x)*y(x)^2 + f1(x)*y(x) + f0(x);
 ${\mathrm{Abel_ode2C}}{≔}\left({\mathrm{g1}}{}\left({x}\right){}{y}{}\left({x}\right){+}{\mathrm{g0}}{}\left({x}\right)\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){=}{\mathrm{f3}}{}\left({x}\right){}{{y}{}\left({x}\right)}^{{3}}{+}{\mathrm{f2}}{}\left({x}\right){}{{y}{}\left({x}\right)}^{{2}}{+}{\mathrm{f1}}{}\left({x}\right){}{y}{}\left({x}\right){+}{\mathrm{f0}}{}\left({x}\right)$ (1)
 where f3(x), f2(x), f1(x), f0(x), g1(x) and g0(x) are arbitrary functions. See Differentialgleichungen, by E. Kamke, p. 28. There is as yet no general solution for this ODE.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (2)

All ODEs of type Abel, second kind, can be rewritten as ODEs of type Abel, first kind, using the following transformation:

 > $\mathrm{with}\left(\mathrm{PDEtools},\mathrm{dchange}\right)$
 $\left[{\mathrm{dchange}}\right]$ (3)
 > $\mathrm{ITR}≔\left\{y\left(x\right)=\frac{1}{u\left(t\right)\mathrm{g1}\left(t\right)}-\frac{\mathrm{g0}\left(t\right)}{\mathrm{g1}\left(t\right)},x=t\right\}$
 ${\mathrm{ITR}}{≔}\left\{{x}{=}{t}{,}{y}{}\left({x}\right){=}\frac{{1}}{{u}{}\left({t}\right){}{\mathrm{g1}}{}\left({t}\right)}{-}\frac{{\mathrm{g0}}{}\left({t}\right)}{{\mathrm{g1}}{}\left({t}\right)}\right\}$ (4)
 > $\mathrm{new_ode}≔\mathrm{dchange}\left(\mathrm{ITR},\mathrm{Abel_ode2C},\left[u\left(t\right),t\right]\right):$
 > $\mathrm{new_ode}≔\mathrm{collect}\left(\frac{ⅆ}{ⅆt}u\left(t\right)=\mathrm{solve}\left(\mathrm{new_ode},\frac{ⅆ}{ⅆt}u\left(t\right)\right),u\left(t\right)\right)$
 ${\mathrm{new_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{u}{}\left({t}\right){=}\frac{\left({\mathrm{f3}}{}\left({t}\right){}{{\mathrm{g0}}{}\left({t}\right)}^{{3}}{-}{{\mathrm{g1}}{}\left({t}\right)}^{{3}}{}{\mathrm{f0}}{}\left({t}\right){+}{{\mathrm{g1}}{}\left({t}\right)}^{{2}}{}{\mathrm{g0}}{}\left({t}\right){}{\mathrm{f1}}{}\left({t}\right){-}{\mathrm{g1}}{}\left({t}\right){}{{\mathrm{g0}}{}\left({t}\right)}^{{2}}{}{\mathrm{f2}}{}\left({t}\right)\right){}{{u}{}\left({t}\right)}^{{3}}}{{{\mathrm{g1}}{}\left({t}\right)}^{{2}}}{+}\frac{\left({-}{3}{}{\mathrm{f3}}{}\left({t}\right){}{{\mathrm{g0}}{}\left({t}\right)}^{{2}}{-}{{\mathrm{g1}}{}\left({t}\right)}^{{2}}{}{\mathrm{f1}}{}\left({t}\right){-}{{\mathrm{g1}}{}\left({t}\right)}^{{2}}{}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{g0}}{}\left({t}\right)\right){+}{2}{}{\mathrm{g1}}{}\left({t}\right){}{\mathrm{g0}}{}\left({t}\right){}{\mathrm{f2}}{}\left({t}\right){+}{\mathrm{g1}}{}\left({t}\right){}{\mathrm{g0}}{}\left({t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{g1}}{}\left({t}\right)\right)\right){}{{u}{}\left({t}\right)}^{{2}}}{{{\mathrm{g1}}{}\left({t}\right)}^{{2}}}{+}\frac{\left({3}{}{\mathrm{f3}}{}\left({t}\right){}{\mathrm{g0}}{}\left({t}\right){-}{\mathrm{g1}}{}\left({t}\right){}{\mathrm{f2}}{}\left({t}\right){-}{\mathrm{g1}}{}\left({t}\right){}\left(\frac{{ⅆ}}{{ⅆ}{t}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{g1}}{}\left({t}\right)\right)\right){}{u}{}\left({t}\right)}{{{\mathrm{g1}}{}\left({t}\right)}^{{2}}}{-}\frac{{\mathrm{f3}}{}\left({t}\right)}{{{\mathrm{g1}}{}\left({t}\right)}^{{2}}}$ (5)
 > $\mathrm{odeadvisor}\left(\mathrm{new_ode},u\left(t\right),\left[\mathrm{Abel}\right]\right)$
 $\left[{\mathrm{_Abel}}\right]$ (6)