Solving ODEs That Are in Quadrature Format - Maple Programming Help

Solving ODEs That Are in Quadrature Format

Description

 • An ODE is said to be in quadrature format when the following conditions are met:
 1) the ODE is of first order and the right hand sides below depend only on x or y(x):
 ${\mathrm{quadrature_1_x_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{F}{}\left({x}\right)$ (1)
 ${\mathrm{quadrature_1_y_ode}}{≔}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{F}{}\left({y}{}\left({x}\right)\right)$ (2)
 2) the ODE is of high order and the right hand side depends only on x. For example:
 ${\mathrm{quadrature_h_x_ode}}{≔}\frac{{{ⅆ}}^{{4}}}{{ⅆ}{{x}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{F}{}\left({x}\right)$ (3)
 where F is an arbitrary function. These ODEs are just integrals in disguised format, and are solved mainly by integrating both sides.

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor},\mathrm{symgen}\right)$
 $\left[{\mathrm{odeadvisor}}{,}{\mathrm{symgen}}\right]$ (4)
 > $\mathrm{odeadvisor}\left(\mathrm{quadrature_1_x_ode}\right)$
 $\left[{\mathrm{_quadrature}}\right]$ (5)
 > $\mathrm{dsolve}\left(\mathrm{quadrature_1_x_ode}\right)$
 ${y}{}\left({x}\right){=}{\int }{F}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}{\mathrm{_C1}}$ (6)
 > $\mathrm{odeadvisor}\left(\mathrm{quadrature_1_y_ode}\right)$
 $\left[{\mathrm{_quadrature}}\right]$ (7)
 > $\mathrm{dsolve}\left(\mathrm{quadrature_1_y_ode}\right)$
 ${x}{-}\left({{\int }}_{{}}^{{y}{}\left({x}\right)}\frac{{1}}{{F}{}\left({\mathrm{_a}}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}\right){+}{\mathrm{_C1}}{=}{0}$ (8)

From the point of view of their symmetries, all ODEs "missing y" have the symmetry [xi = 0, eta = 1], and all ODEs "missing x" have the symmetry [xi = 1, eta = 0] (see symgen);

 > $\mathrm{symgen}\left(\mathrm{quadrature_1_x_ode}\right)$
 $\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}{1}\right]$ (9)
 > $\mathrm{symgen}\left(\mathrm{quadrature_1_y_ode}\right)$
 $\left[{\mathrm{_ξ}}{=}{1}{,}{\mathrm{_η}}{=}{0}\right]$ (10)