More AP Calculus Integration Homework Help!?

Question

Consider the differential equation
dy 3x²
---- = ------
dx e²*

(because of my browser, the * is supposed to be a "y")

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

b) find the domain and range of the function "f" found in part a.

Answer 1

dy/dx = 3x^2 / e^2y
e^2y dy = 3x^2 dx

Integrate both sides:
∫e^2y dy = 2e^2y + c
∫3x^2 dx = x^3 + c

e^2y = x^3 + c
e^2y = x^3 + c
ln e^2y = ln (x^3 + c)
2y = ln (x^3 + c): ln b^a = a ln b
y = ln (x^3 + c) / 2

a.) plug 0 in:
ln (0 + c) / 2 = 1/2
ln c = 1
c = e
y = ln (x^3 + e) / 2

domain = x > -e^1/3 (you can't take the log of a negative number or 0)
range = all real numbers