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Solving ODEs That Do Not Contain Either the Dependent or Independent Variable

Description

 • The general form of an nth order ODE that is missing the dependent variable is:
 > missing_y_ode := F(x,'seq(diff(y(x),x$i),i=1..n)');  ${\mathrm{missing_y_ode}}{≔}{F}{}\left({x}{,}{\mathrm{seq}}{}\left(\frac{{{ⅆ}}^{{i}}}{{ⅆ}{{x}}^{{i}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){,}{i}{=}{1}{..}{n}\right)\right)$ (1)  where F is an arbitrary function of its arguments. The order can be reduced by introducing a new variable p(x) = diff(y(x),x). If the reduced ODE can be solved for p(x), the solution to the original ODE is determined as a quadrature.  • The general form of an nth order ODE that is missing the independent variable is:  > missing_x_ode := F(y(x),'seq(diff(y(x),x$i),i=1..n)');
 ${\mathrm{missing_x_ode}}{≔}{F}{}\left({y}{}\left({x}\right){,}{\mathrm{seq}}{}\left(\frac{{{ⅆ}}^{{i}}}{{ⅆ}{{x}}^{{i}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){,}{i}{=}{1}{..}{n}\right)\right)$ (2)
 where F is an arbitrary function of its arguments. The transformation $y\text{'}=p,y\mathrm{\text{'}\text{'}}=p\mathrm{p\text{'}},y\mathrm{\text{'}\text{'}\text{'}}={p}^{2}\mathrm{p\text{'}\text{'}}+p{\left(\mathrm{p\text{'}}\right)}^{2},\mathrm{...}$
 yields a reduction of order. If the reduced ODE can be solved for p(y), the solution to the original ODE can be given implicitly as
 > x = Int(1/p(y),y) + _C1;
 ${x}{=}{\int }\frac{{1}}{{p}{}\left({y}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{y}{+}{\mathrm{_C1}}$ (3)
 See Murphy, "Ordinary Differential Equations and their Solutions", 1960, sections B2(1,2), and C2(1,2).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools},\mathrm{odeadvisor}\right)$
 $\left[{\mathrm{odeadvisor}}\right]$ (4)
 > $\mathrm{_2nd_order_missing_x_ode}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)=\left(\mathrm{ln}\left(y\left(x\right)\right)+1\right)\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)$
 ${\mathrm{_2nd_order_missing_x_ode}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}\left({\mathrm{ln}}{}\left({y}{}\left({x}\right)\right){+}{1}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (5)
 > $\mathrm{odeadvisor}\left(\mathrm{_2nd_order_missing_x_ode}\right)$
 $\left[\left[{\mathrm{_2nd_order}}{,}{\mathrm{_missing_x}}\right]{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_exact}}{,}{\mathrm{_nonlinear}}\right]{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_reducible}}{,}{\mathrm{_mu_x_y1}}\right]{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_reducible}}{,}{\mathrm{_mu_xy}}\right]\right]$ (6)
 > ${\mathrm{sol}}_{\mathrm{x2}}≔\mathrm{dsolve}\left(\mathrm{_2nd_order_missing_x_ode}\right)$
 ${{\mathrm{sol}}}_{{\mathrm{x2}}}{≔}{y}{}\left({x}\right){=}\mathrm{c__1}{,}{{\int }}_{{}}^{{y}{}\left({x}\right)}\frac{{1}}{{\mathrm{_a}}{}{\mathrm{ln}}{}\left({\mathrm{_a}}\right){+}\mathrm{c__1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{-}{x}{-}\mathrm{c__2}{=}{0}$ (7)

Explicit and implicit answers can be tested, in principle, using odetest:

 > $\mathrm{map}\left(\mathrm{odetest},\left[{\mathrm{sol}}_{\mathrm{x2}}\right],\mathrm{_2nd_order_missing_x_ode}\right)$
 $\left[{0}{,}{0}\right]$ (8)
 > $\mathrm{_2nd_order_missing_y_ode}≔\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)=F\left(x\right){\left(\frac{ⅆ}{ⅆx}y\left(x\right)\right)}^{3}$
 ${\mathrm{_2nd_order_missing_y_ode}}{≔}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{F}{}\left({x}\right){}{\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)}^{{3}}$ (9)
 > $\mathrm{odeadvisor}\left(\mathrm{_2nd_order_missing_y_ode}\right)$
 $\left[\left[{\mathrm{_2nd_order}}{,}{\mathrm{_missing_y}}\right]{,}\left[{\mathrm{_2nd_order}}{,}{\mathrm{_reducible}}{,}{\mathrm{_mu_y_y1}}\right]\right]$ (10)
 > ${\mathrm{sol}}_{\mathrm{y2}}≔\mathrm{dsolve}\left(\mathrm{_2nd_order_missing_y_ode}\right)$
 ${{\mathrm{sol}}}_{{\mathrm{y2}}}{≔}{y}{}\left({x}\right){=}{\int }\frac{{1}}{\sqrt{\mathrm{c__1}{-}{2}{}\left({\int }{F}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}\mathrm{c__2}{,}{y}{}\left({x}\right){=}{\int }{-}\frac{{1}}{\sqrt{\mathrm{c__1}{-}{2}{}\left({\int }{F}{}\left({x}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}{+}\mathrm{c__2}$ (11)

In the case of multiple answers it is convenient to "map" odetest as follows:

 > $\mathrm{map}\left(\mathrm{odetest},\left[{\mathrm{sol}}_{\mathrm{y2}}\right],\mathrm{_2nd_order_missing_y_ode}\right)$
 $\left[{0}{,}{0}\right]$ (12)

The most general third order ODE missing x. This ODE cannot be solved to the end: its solution involves the solving of the most general second order ODE. However, its differential order can be reduced (see ?dsolve,ODESolStruc):

 > $\mathrm{_3rd_order_missing_x_ode}≔\frac{{ⅆ}^{3}}{ⅆ{x}^{3}}y\left(x\right)=F\left(y\left(x\right),\frac{ⅆ}{ⅆx}y\left(x\right),\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)$
 ${\mathrm{_3rd_order_missing_x_ode}}{≔}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{F}{}\left({y}{}\left({x}\right){,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){,}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (13)
 > ${\mathrm{sol}}_{\mathrm{x3}}≔\mathrm{dsolve}\left(\mathrm{_3rd_order_missing_x_ode}\right)$
 ${{\mathrm{sol}}}_{{\mathrm{x3}}}{≔}{y}{}\left({x}\right){=}{\mathrm{_a}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left[\left\{\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{_a}}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\right){}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}^{{2}}{+}{\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\right)}^{{2}}{}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){-}{F}{}\left({\mathrm{_a}}{,}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){,}\left(\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\right){}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\right){=}{0}\right\}{,}\left\{{\mathrm{_a}}{=}{y}{}\left({x}\right){,}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right\}{,}\left\{{x}{=}{\int }\frac{{1}}{{\mathrm{_b}}{}\left({\mathrm{_a}}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}\mathrm{c__1}{,}{y}{}\left({x}\right){=}{\mathrm{_a}}\right\}\right]$ (14)
 > $\mathrm{odeadvisor}\left(\mathrm{_3rd_order_missing_x_ode}\right)$
 $\left[\left[{\mathrm{_3rd_order}}{,}{\mathrm{_missing_x}}\right]{,}\left[{\mathrm{_3rd_order}}{,}{\mathrm{_with_linear_symmetries}}\right]\right]$ (15)
 > $\mathrm{odetest}\left({\mathrm{sol}}_{\mathrm{x3}},\mathrm{_3rd_order_missing_x_ode}\right)$
 ${0}$ (16)

The most general third order ODE missing y.

 > $\mathrm{_3rd_order_missing_y_ode}≔\frac{{ⅆ}^{3}}{ⅆ{x}^{3}}y\left(x\right)=F\left(x,\frac{ⅆ}{ⅆx}y\left(x\right),\frac{{ⅆ}^{2}}{ⅆ{x}^{2}}y\left(x\right)\right)$
 ${\mathrm{_3rd_order_missing_y_ode}}{≔}\frac{{{ⅆ}}^{{3}}}{{ⅆ}{{x}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){=}{F}{}\left({x}{,}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right){,}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right)$ (17)
 > $\mathrm{odeadvisor}\left(\mathrm{_3rd_order_missing_y_ode}\right)$
 $\left[\left[{\mathrm{_3rd_order}}{,}{\mathrm{_missing_y}}\right]{,}\left[{\mathrm{_3rd_order}}{,}{\mathrm{_with_linear_symmetries}}\right]\right]$ (18)
 > ${\mathrm{sol}}_{\mathrm{y3}}≔\mathrm{dsolve}\left(\mathrm{_3rd_order_missing_y_ode}\right)$
 ${{\mathrm{sol}}}_{{\mathrm{y3}}}{≔}{y}{}\left({x}\right){=}\left({\int }{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}\mathrm{c__1}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left[\left\{\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{\mathrm{_a}}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}{F}{}\left({\mathrm{_a}}{,}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){,}\frac{{ⅆ}}{{ⅆ}{\mathrm{_a}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\right)\right\}{,}\left\{{\mathrm{_a}}{=}{x}{,}{\mathrm{_b}}{}\left({\mathrm{_a}}\right){=}\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right\}{,}\left\{{x}{=}{\mathrm{_a}}{,}{y}{}\left({x}\right){=}{\int }{\mathrm{_b}}{}\left({\mathrm{_a}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{_a}}{+}\mathrm{c__1}\right\}\right]$ (19)
 > $\mathrm{odetest}\left({\mathrm{sol}}_{\mathrm{y3}},\mathrm{_3rd_order_missing_y_ode}\right)$
 ${0}$ (20)