Q:

Transforming differential equations

I feel as though there is a simple answer to this question but I can't seem to understand what was done, my lecturer has an example that begins like so: 
$$x^2\frac{d^2y}{dx^2}+y=0,$$ then if we let $y(x) = w(t)$ with $x = \frac{1}{t},$ then our equation becomes  $$ \frac{1}{t^2}(t^4\frac{d^2w}{dt^2} + 2t^3\frac{dw}{dt}) + w = 0.$$
I guess this is just some basic use of substitution and chain rule or product rule maybe? I don't really understand how we find the part in the bracket and apologise if it is very simple but could someone clarify for me! 

A:

Introducing $w$ to replace the $y$ seems unnecessary, but nevertheless, 
$$\frac {dy}{dx}=\frac{dw}{dx}=\frac{dw}{dt}\cdot\frac{dt}{dx}$$ using the Chain Rule.
Meanwhile, $$\frac{dx}{dt}=-\frac{1}{t^2}$$
$$\implies\frac{dy}{dx}=-t^2\frac{dw}{dt}$$
Now $$\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dt}\left(-t^2\frac{dw}{dt}\right)\frac{dt}{dx}$$ using the Chain Rule again.
Now use the product rule to differentiate the terms in the bracket and the result will follow immediately.
I hope this helps.

