If, for an nth order ODE (n=2 or n=3) with the nth derivative isolated, there exists an integrating factor which depends only on the (n-1)st derivative, this integrating factor can be determined. The differential order of the ODE can then be reduced by one.

The general form of such an ODE of second order is:

reducible_ode_2 :=
diff(y(x),x,x)=diff(G(x,y(x)),x)/D(F)(diff(y(x),x));

$\mathrm{reducible\_ode\_2}≔\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)=\frac{{\mathrm{D}}_1\left(G\right)\left(x\,y\left(x\right)\right)+{\mathrm{D}}_2\left(G\right)\left(x\,y\left(x\right)\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}{\mathrm{D}\left(F\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}$

where F and G are arbitrary functions of their arguments. The integrating factor in this case is

mu := D(F)(diff(y(x),x));

$\mathrm{\mu }≔\mathrm{D}\left(F\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)$

The reduced ODE then becomes

F(diff(y(x),x)) = G(x,y(x)) + _C1;

$F\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)=G\left(x\,y\left(x\right)\right)+\mathrm{\_C1}$

The general form of this ODE of third order is:

reducible_ode_3 :=
diff(y(x),x$3)=diff(G(x,y(x),diff(y(x),x)),x)/D(F)(diff(y(x),x,x));

$\mathrm{reducible\_ode\_3}≔\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)=\frac{{\mathrm{D}}_1\left(G\right)\left(x\,y\left(x\right)\,\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+{\mathrm{D}}_2\left(G\right)\left(x\,y\left(x\right)\,\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+{\mathrm{D}}_3\left(G\right)\left(x\,y\left(x\right)\,\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}{\mathrm{D}\left(F\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}$

where F and G are arbitrary functions of their arguments. The integrating factor in this case is

mu := D(F)(diff(y(x),x,x));

$\mathrm{\mu }≔\mathrm{D}\left(F\right)\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)$

The reduced ODE is

F(diff(y(x),x,x)) = G(x,y(x),diff(y(x),x)) + _C1;

$F\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)=G\left(x\,y\left(x\right)\,\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\mathrm{\_C1}$

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

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

$\mathrm{ode}\left[1\right]≔\frac{x\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+2\mathrm{diff}\left(y\left(x\right)\,x\right)}{x^2}-\frac{x\mathrm{diff}\left(y\left(x\right)\,x\,x\right)+2\mathrm{diff}\left(y\left(x\right)\,x\right)}{x^2{y\left(x\right)}^2}+\frac{\frac{2}{x}{\mathrm{diff}\left(y\left(x\right)\,x\right)}^2}{{y\left(x\right)}^3}=0$

${\mathrm{ode}}_1≔\frac{x\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+2\frac{\ⅆ}{\ⅆx}y\left(x\right)}{x^2}-\frac{x\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)+2\frac{\ⅆ}{\ⅆx}y\left(x\right)}{x^2{y\left(x\right)}^2}+\frac{2{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)}^2}{x{y\left(x\right)}^3}=0$

$\mathrm{odeadvisor}\left(\mathrm{ode}\left[1\right]\right)$

$\left[\mathrm{\_Liouville}\,\left[\mathrm{\_2nd\_order}\,\mathrm{\_with\_linear\_symmetries}\right]\,\left[\mathrm{\_2nd\_order}\,\mathrm{\_reducible}\,\mathrm{\_mu\_x\_y1}\right]\,\left[\mathrm{\_2nd\_order}\,\mathrm{\_reducible}\,\mathrm{\_mu\_xy}\right]\right]$

$\mathrm{sol}≔\mathrm{dsolve}\left(\mathrm{ode}\left[1\right]\right)$

$\mathrm{sol}≔y\left(x\right)=\frac{\mathrm{c\_\_2}x-\mathrm{c\_\_1}+\sqrt{{\mathrm{c\_\_2}}^2{x}^2-2\mathrm{c\_\_1}\mathrm{c\_\_2}x+{\mathrm{c\_\_1}}^2-4x^2}}{2x},y\left(x\right)=-\frac{-\mathrm{c\_\_2}x+\sqrt{{\mathrm{c\_\_2}}^2{x}^2-2\mathrm{c\_\_1}\mathrm{c\_\_2}x+{\mathrm{c\_\_1}}^2-4x^2}+\mathrm{c\_\_1}}{2x}$

$\mathrm{map}\left(\mathrm{odetest}\,\left[\mathrm{sol}\right]\,\mathrm{ode}\left[1\right]\right)$

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

$\mathrm{ode}\left[2\right]≔\frac{\frac{1}{x}\mathrm{diff}\left(y\left(x\right)\,x\,x\,x\right)}{\mathrm{diff}\left(y\left(x\right)\,x\,x\right)}=\frac{\frac{1}{x^2}\left(\mathrm{diff}\left(y\left(x\right)\,x\right)x+y\left(x\right)\right)}{y\left(x\right)}$

${\mathrm{ode}}_2≔\frac{\frac{{\ⅆ}^3}{\ⅆx^3}y\left(x\right)}{x\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)}=\frac{\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)x+y\left(x\right)}{x^{2}y\left(x\right)}$

$\mathrm{odeadvisor}\left(\mathrm{ode}\left[2\right]\right)$

$\left[\left[\mathrm{\_3rd\_order}\,\mathrm{\_with\_linear\_symmetries}\right]\,\left[\mathrm{\_3rd\_order}\,\mathrm{\_reducible}\,\mathrm{\_mu\_y2}\right]\,\left[\mathrm{\_3rd\_order}\,\mathrm{\_reducible}\,\mathrm{\_mu\_poly\_yn}\right]\right]$

$\mathrm{sol}≔\mathrm{dsolve}\left(\mathrm{ode}\left[2\right]\right)$

$\mathrm{sol}≔y\left(x\right)=\mathrm{c\_\_2}\mathrm{AiryAi}\left(-\mathrm{c\_\_1}x\right)+\mathrm{c\_\_3}\mathrm{AiryBi}\left(-\mathrm{c\_\_1}x\right)$