antiderivative of Y'= 3x^2/e^(2y)?

Question

Y' = 3x^2 / e^(2y)

Find a solution y=f(x) to the differential equation satisfying f(0) = 1/2

Find the domain and range of the function f found in the part above

Answer 1

Separating variables, you have
e^2ydy = 3x^2dx
(1/2)e^2y = x^3 + C
e^2y = 2x^3 + C
2y = Ln(2x^3 + C)
y = (1/2)Ln(2x^3 + C)
y(0) = 1/2 = (1/2)Ln(0 + C)
C = e
y = (1/2)Ln(2x^3 + e)

domain is x > - (e/2)^1/3
range is all y