Question on transformation of second order ODE

Question

In many literature related to ODE BVP, people write a second order ODE

\begin{equation}y''=f_0(x,y,y')\end{equation}

as

\begin{equation}y''=f(x,y)\end{equation}

It seems that the first equation can be transformed to the second form, but I don't know.

(Question) Is there a general transformation from the first one to the second one?

Answer 1

The general answer is: no, there is no general transformation to remove the dependence on $y'$.

However, a large number of second order boundary value problems studied in the literature are linear, i.e. of Sturm-Liouville type: $$ y'' + a_1(x) y' + a_2(x) y = f(x). \tag{1} $$ In these cases, you can apply the Liouville transform $$ y(x) = z(x) \text{exp}\,\left[ \int^x -\frac{1}{2} a_1(\xi)\,\text{d}\xi\right] $$ which transforms $(1)$ into $$ z'' + b(x) z = f(x)\,\text{exp}\left[ \int^x \frac{1}{2} a_1(\xi)\,\text{d}\xi\right], $$ where $$ b(x) = -\frac{1}{2} a_1' - \frac{1}{4} a_1^2 + a_2. $$