(dy/dx)^2=(x^2/y)?

Question

(dy/dx)^2=(x^2/y)
these powers are powers
not derivatives
so its the 2nd derivative=x squared over y

the 2nd dertivative is the same as y''/x''

Answer 1

So, solving 2nd order diff equations; well I will give you a little help - considering the date this can't be from a DE class , must be calc or ...

you can view this

y'' = x^2 / y

d/dx (y') = x^2 / y

d (y') = x^2 dx / y

y d (y') = x^2 dx

Now, integrate both sides [don't forget constants due to indefinite integration and avoid pitfalls :) ]

the rhs is trivial
i haven't bothered to think much about the lhs nor am i going to as i have some work to do ; might try integration by parts or i guess mathematica or maple , whatever you do if you end up with a y ' still then you'll have to do another step and then solve for y

Answer 2

If (dy/dx)² = (x²/y)
then;
dy²/dx² = x²/y eq.1
dy² = (dx²) x²/y eq.2
Now substitute eq.2 into eq.1.
[(dx²) x²/y] /dx² = x²/y
x²/y = x²/y
→
(dy/dx)² = (x²/y)