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)');

$\mathrm{missing\_y\_ode}≔F\left(x\,\mathrm{seq}\left(\frac{{\ⅆ}^i}{\ⅆx^i}y\left(x\right)\,i=1..n\right)\right)$

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)');

$\mathrm{missing\_x\_ode}≔F\left(y\left(x\right)\,\mathrm{seq}\left(\frac{{\ⅆ}^i}{\ⅆx^i}y\left(x\right)\,i=1..n\right)\right)$

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;

$x=\int \frac{1}{p\left(y\right)}\ⅆy+\mathrm{\_C1}$

See Murphy, "Ordinary Differential Equations and their Solutions", 1960, sections B2(1,2), and C2(1,2).

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\right)$

$\left[\mathrm{odeadvisor}\right]$

$\mathrm{\_2nd\_order\_missing\_x\_ode}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=\left(\ln \left(y\left(x\right)\right)+1\right)\mathrm{diff}\left(y\left(x\right)\,x\right)$

$\mathrm{\_2nd\_order\_missing\_x\_ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\left(\ln \left(y\left(x\right)\right)+1\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)$

$\mathrm{odeadvisor}\left(\mathrm{\_2nd\_order\_missing\_x\_ode}\right)$

$\left[\left[\mathrm{\_2nd\_order}\,\mathrm{\_missing\_x}\right]\,\left[\mathrm{\_2nd\_order}\,\mathrm{\_exact}\,\mathrm{\_nonlinear}\right]\,\left[\mathrm{\_2nd\_order}\,\mathrm{\_reducible}\,\mathrm{\_mu\_x\_y1}\right]\,\left[\mathrm{\_2nd\_order}\,\mathrm{\_reducible}\,\mathrm{\_mu\_xy}\right]\right]$

$\mathrm{sol}\left[\mathrm{x2}\right]≔\mathrm{dsolve}\left(\mathrm{\_2nd\_order\_missing\_x\_ode}\right)$

${\mathrm{sol}}_{\mathrm{x2}}≔y\left(x\right)=\mathrm{c\_\_1},{\int }_{\phantom{\mathrm{`\; `}}}^{y\left(x\right)}\frac{1}{\mathrm{\_a}\ln \left(\mathrm{\_a}\right)+\mathrm{c\_\_1}}\ⅆ\mathrm{\_a}-x-\mathrm{c\_\_2}=0$

$\mathrm{map}\left(\mathrm{odetest}\,\left[\mathrm{sol}\left[\mathrm{x2}\right]\right]\,\mathrm{\_2nd\_order\_missing\_x\_ode}\right)$

$\left[0\,0\right]$

$\mathrm{\_2nd\_order\_missing\_y\_ode}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\right)=F\left(x\right){\mathrm{diff}\left(y\left(x\right)\,x\right)}^3$

$\mathrm{\_2nd\_order\_missing\_y\_ode}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=F\left(x\right){\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^3$

$\mathrm{odeadvisor}\left(\mathrm{\_2nd\_order\_missing\_y\_ode}\right)$

$\left[\left[\mathrm{\_2nd\_order}\,\mathrm{\_missing\_y}\right]\,\left[\mathrm{\_2nd\_order}\,\mathrm{\_reducible}\,\mathrm{\_mu\_y\_y1}\right]\right]$

$\mathrm{sol}\left[\mathrm{y2}\right]≔\mathrm{dsolve}\left(\mathrm{\_2nd\_order\_missing\_y\_ode}\right)$

${\mathrm{sol}}_{\mathrm{y2}}≔y\left(x\right)=\int \frac{1}{\sqrt{\mathrm{c\_\_1}-2\int F\left(x\right)\ⅆx}}\ⅆx+\mathrm{c\_\_2},y\left(x\right)=\int -\frac{1}{\sqrt{\mathrm{c\_\_1}-2\int F\left(x\right)\ⅆx}}\ⅆx+\mathrm{c\_\_2}$

$\mathrm{map}\left(\mathrm{odetest}\,\left[\mathrm{sol}\left[\mathrm{y2}\right]\right]\,\mathrm{\_2nd\_order\_missing\_y\_ode}\right)$

$\left[0\,0\right]$

$\mathrm{\_3rd\_order\_missing\_x\_ode}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)=F\left(y\left(x\right)\,\mathrm{diff}\left(y\left(x\right)\,x\right)\,\mathrm{diff}\left(y\left(x\right)\,x\,x\right)\right)$

$\mathrm{\_3rd\_order\_missing\_x\_ode}≔\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)=F\left(y\left(x\right)\,\frac{\ⅆ}{\ⅆx}y\left(x\right)\,\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

$\mathrm{sol}\left[\mathrm{x3}\right]≔\mathrm{dsolve}\left(\mathrm{\_3rd\_order\_missing\_x\_ode}\right)$

${\mathrm{sol}}_{\mathrm{x3}}≔y\left(x\right)=\mathrm{\_a}\mathbf{where}\left[\left\{\left(\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\_a}}^2}\mathrm{\_b}\left(\mathrm{\_a}\right)\right){\mathrm{\_b}\left(\mathrm{\_a}\right)}^2+{\left(\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)\right)}^2\mathrm{\_b}\left(\mathrm{\_a}\right)-F\left(\mathrm{\_a}\,\mathrm{\_b}\left(\mathrm{\_a}\right)\,\left(\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)\right)\mathrm{\_b}\left(\mathrm{\_a}\right)\right)=0\right\}\,\left\{\mathrm{\_a}=y\left(x\right)\,\mathrm{\_b}\left(\mathrm{\_a}\right)=\frac{\ⅆ}{\ⅆx}y\left(x\right)\right\}\,\left\{x=\int \frac{1}{\mathrm{\_b}\left(\mathrm{\_a}\right)}\ⅆ\mathrm{\_a}+\mathrm{c\_\_1}\,y\left(x\right)=\mathrm{\_a}\right\}\right]$

$\mathrm{odeadvisor}\left(\mathrm{\_3rd\_order\_missing\_x\_ode}\right)$

$\left[\left[\mathrm{\_3rd\_order}\,\mathrm{\_missing\_x}\right]\,\left[\mathrm{\_3rd\_order}\,\mathrm{\_with\_linear\_symmetries}\right]\right]$

$\mathrm{odetest}\left(\mathrm{sol}\left[\mathrm{x3}\right]\,\mathrm{\_3rd\_order\_missing\_x\_ode}\right)$

$0$

$\mathrm{\_3rd\_order\_missing\_y\_ode}≔\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)=F\left(x\,\mathrm{diff}\left(y\left(x\right)\,x\right)\,\mathrm{diff}\left(y\left(x\right)\,x\,x\right)\right)$

$\mathrm{\_3rd\_order\_missing\_y\_ode}≔\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)=F\left(x\,\frac{\ⅆ}{\ⅆx}y\left(x\right)\,\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

$\mathrm{odeadvisor}\left(\mathrm{\_3rd\_order\_missing\_y\_ode}\right)$

$\left[\left[\mathrm{\_3rd\_order}\,\mathrm{\_missing\_y}\right]\,\left[\mathrm{\_3rd\_order}\,\mathrm{\_with\_linear\_symmetries}\right]\right]$

$\mathrm{sol}\left[\mathrm{y3}\right]≔\mathrm{dsolve}\left(\mathrm{\_3rd\_order\_missing\_y\_ode}\right)$

${\mathrm{sol}}_{\mathrm{y3}}≔y\left(x\right)=\left(\int \mathrm{\_b}\left(\mathrm{\_a}\right)\ⅆ\mathrm{\_a}+\mathrm{c\_\_1}\right)\mathbf{where}\left[\left\{\frac{{\ⅆ}^2}{\ⅆ{\mathrm{\_a}}^2}\mathrm{\_b}\left(\mathrm{\_a}\right)=F\left(\mathrm{\_a}\,\mathrm{\_b}\left(\mathrm{\_a}\right)\,\frac{\ⅆ}{\ⅆ\mathrm{\_a}}\mathrm{\_b}\left(\mathrm{\_a}\right)\right)\right\}\,\left\{\mathrm{\_a}=x\,\mathrm{\_b}\left(\mathrm{\_a}\right)=\frac{\ⅆ}{\ⅆx}y\left(x\right)\right\}\,\left\{x=\mathrm{\_a}\,y\left(x\right)=\int \mathrm{\_b}\left(\mathrm{\_a}\right)\ⅆ\mathrm{\_a}+\mathrm{c\_\_1}\right\}\right]$

$\mathrm{odetest}\left(\mathrm{sol}\left[\mathrm{y3}\right]\,\mathrm{\_3rd\_order\_missing\_y\_ode}\right)$

$0$