exact nonlinear - Maple Help
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Exact Nonlinear ODEs

Description

 • The general form of the exact nonlinear ODE is given by the following:
 > exact_nonlinear_ode := 'diff(F(x,y(x),seq(diff(y(x),x$i),i=1..n)),x)' = 0;  ${\mathrm{exact_nonlinear_ode}}{≔}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{F}{}\left({x}{,}{y}{}\left({x}\right){,}{\mathrm{seq}}{}\left(\frac{{{ⅆ}}^{{i}}}{{ⅆ}{{x}}^{{i}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){,}{i}{=}{1}{..}{n}\right)\right){=}{0}$ (1)  See Murphy, "Ordinary Differential Equations and their Solutions", p. 221.  • The order of this ODE can be reduced since it is the total derivative of an ODE of one order lower. If the given ODE is G(x,y,y1,y2,...,yn)=0, the test for exactness is the following: $\mathrm{g0}-\mathrm{D}\left(\mathrm{g1}\right)+{\mathrm{D}}^{2}\left(\mathrm{g2}\right)-\cdots ±{\mathrm{D}}^{n}\left(\mathrm{gn}\right)=0$  where $\mathrm{D}=\frac{ⅆ}{ⅆx},\left(\mathrm{y1},\dots ,\mathrm{yn}\mathrm{being}\mathrm{functions}\mathrm{of}x\right)$ $\mathrm{gn}={\mathrm{D}}_{n+1}\left(G\right)\left(x,y,\mathrm{y1},\mathrm{y2},\mathrm{...},\mathrm{yn}\right)\left(=\frac{ⅆG}{ⅆ\mathrm{yn}}\right),$ $\mathrm{yn}=\frac{{ⅆ}^{n}}{{ⅆx}^{n}}y\left(x\right)$  Note: The derivatives with respect to y, dy/dx and d^2y/dx^2 are taken in the obvious manner but the derivatives with regard to x are taken considering y, and its derivatives as functions of x.  The reduced ODE is:  > reduced_ode := 'F(x,y(x),seq(diff(y(x),x$i),i=1..n))' = _C1;
 ${\mathrm{reduced_ode}}{≔}{F}{}\left({x}{,}{y}{}\left({x}\right){,}{\mathrm{seq}}{}\left(\frac{{{ⅆ}}^{{i}}}{{ⅆ}{{x}}^{{i}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){,}{i}{=}{1}{..}{n}\right)\right){=}{\mathrm{_C1}}$ (2)

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (3)
 > $\mathrm{ode}≔\mathrm{diff}\left(y\left(x\right),x,x\right)=\frac{1}{y\left(x\right)}-\frac{x}{{y\left(x\right)}^{2}}\mathrm{diff}\left(y\left(x\right),x\right)$
 ${\mathrm{ode}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\frac{{1}}{{y}{}\left({x}\right)}{-}\frac{{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}{{{y}{}\left({x}\right)}^{{2}}}$ (4)
 > $\mathrm{odeadvisor}\left(\mathrm{ode}\right)$
 $\left[\left[{\mathrm{_2nd_order}}{,}{\mathrm{_exact}}{,}{\mathrm{_nonlinear}}\right]{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_with_linear_symmetries}}\right]{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_reducible}}{,}{\mathrm{_mu_x_y1}}\right]{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_reducible}}{,}{\mathrm{_mu_y_y1}}\right]{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_reducible}}{,}{\mathrm{_mu_xy}}\right]\right]$ (5)
 > $\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{ode},\mathrm{implicit}\right)$
 ${\mathrm{ans}}{≔}{-}\frac{{\mathrm{ln}}{}\left(\frac{\mathrm{c__1}{}{x}{}{y}{}\left({x}\right){-}{{x}}^{{2}}{+}{{y}{}\left({x}\right)}^{{2}}}{{{x}}^{{2}}}\right)}{{2}}{-}\frac{\mathrm{c__1}{}{\mathrm{arctanh}}{}\left(\frac{\mathrm{c__1}{}{x}{+}{2}{}{y}{}\left({x}\right)}{{x}{}\sqrt{{\mathrm{c__1}}^{{2}}{+}{4}}}\right)}{\sqrt{{\mathrm{c__1}}^{{2}}{+}{4}}}{-}{\mathrm{ln}}{}\left({x}\right){-}\mathrm{c__2}{=}{0}$ (6)
 > $\mathrm{odetest}\left(\mathrm{ans},\mathrm{ode}\right)$
 ${0}$ (7)