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));

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

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

The reduced ODE then becomes

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

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));

The reduced ODE is

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