Undergrad Integral-form change of variable in differential equation

[Jaime_mc2]

I have the following differential equation, which is the general Sturm-Liouville problem,
$$
\dfrac{d}{dx} \left[ p(x) \dfrac{d\varphi}{dx} \right] + \left[ \lambda w(x) - q(x) \right] \varphi(x) = 0\ ,
$$
and I want to perform the change of variable
$$
x \rightarrow y = \int_a^x \sqrt{\lambda \dfrac{w(t)}{p(t)}}\, dt\ .
$$

I used the fundamental theorem of calculus to get
$$
\dfrac{dy}{dx} = \sqrt{\lambda\dfrac{w(x)}{p(x)}}\ ,
$$
and get the differentiation operator with respect to the new variable as
$$
\dfrac{d}{dx} = \dfrac{dy}{dx}\dfrac{d}{dy} = \sqrt{\lambda\dfrac{w(x)}{p(x)}} \dfrac{d}{dy}\ .
$$

My question is about how do I get the differential equation to be expressed in terms of $y$. My first thought was realising that $\varphi(x)$ could just be expressed as $\varphi(y)$ if there exists an inversion such that $x = x(y)$. Then I just extended this idea to $w$, $p$ and $q$, and I arrived to the equation
$$
\dfrac{d}{dy} \left[ \sqrt{w(y)p(y)} \dfrac{d\varphi}{dy} \right] + \left[ \sqrt{w(y)p(y)} - \dfrac{q(y)}{\lambda} \sqrt{\dfrac{p(y)}{w(y)}} \right] \varphi(y) = 0\ .
$$

Is this approach correct?

 

[anuttarasammyak]

I observe if you choose w as w=q, the equation becomes easy enough to solve.