Question

Using the separation of variables technique, solve the following differential equation with initial condition:
(4x sqrt)(1 - t^2)(dx/dt) - 1 = 0 and x(0)=-2
I believe that the answer is one of the following two options:
a.) 2x^2 = arcsint + 8
b.) arccost + 8 - (1/2)pi

Answer 1

4 x dx = dt/sqrt (1-t^2)
2 x^2 = sin^-1 (t) + c
when t = 0, x = 2
8 = 0 + c
c = 8
so 2 x2 = sin^-1 (t) + 8
or
2 x^2 = cos^-1(t) + c
8 = pi/2 + c
yes, agree

Answer 2

I'm only permitted to choose one option. And seeing as you came up with arcsin first, would this be a better answer over the option with arccos?

Answer 3

They are the same.
cos (pi/2 - x) = sin x