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;

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

with(DEtools, odeadvisor);

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

ode[1] := (x*diff(y(x),x,x)+2*diff(y(x),x))/x^2-(x*diff(y(x),x,x)
+2*diff(y(x),x))/x^2/y(x)^2+2/x*diff(y(x),x)^2/y(x)^3 = 0;

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

odeadvisor(ode[1]);

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

sol := dsolve(ode[1]);

$\mathrm{sol}≔y\left(x\right)=\frac{\mathrm{\_C2}x-\mathrm{\_C1}+\sqrt{{\mathrm{\_C2}}^2{x}^2-2\mathrm{\_C1}\mathrm{\_C2}x+{\mathrm{\_C1}}^2-4x^2}}{2x},y\left(x\right)=-\frac{-\mathrm{\_C2}x+\sqrt{{\mathrm{\_C2}}^2{x}^2-2\mathrm{\_C1}\mathrm{\_C2}x+{\mathrm{\_C1}}^2-4x^2}+\mathrm{\_C1}}{2x}$

map(odetest,[sol],ode[1]);

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

ode[2] := 1/x*diff(y(x),x,x,x)/diff(y(x),x,x)=1/x^2*(diff(y(x),x)*x+y(x))/y(x);

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

odeadvisor(ode[2]);

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

sol := dsolve(ode[2]);

$\mathrm{sol}≔y\left(x\right)=\mathrm{\_C2}\mathrm{AiryAi}\left(-\mathrm{\_C1}x\right)+\mathrm{\_C3}\mathrm{AiryBi}\left(-\mathrm{\_C1}x\right)$