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)');

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)');

where F is an arbitrary function of its arguments. The transformation

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;

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