Expert: Ahmed Salami - 1/30/2010

Question
Sorry for your time!

Consider the differential equation dy/dx= (3x^2)/(e^[2y]).

a) Find a solution y=f(x) to the differential equation satisfying f(0)= 0.5 (one half)

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

Thanks so much for any help!

Answer
Hi Devi,
dy/dx = 3x²/e^(2y)
Cross multiplying,
e^(2y) dy = 3x² dx
Integrating bothsides,
∫e^(2y) dy = ∫3x² dx
(1/2)e^(2y) = x³ + c
e^(2y) = 2(x³ + c)
Taking natural logarithm of both sides,
2y = ln 2(x³ + c)
y = (1/2)[ln 2(x³ + c)]
To determine c, we use the initial condition f(0) = 1/2
1/2 = (1/2)[ln 2(0³ + c)]
1/2 = (1/2)[ln(2c)]
1 = ln(2c)
2c = e
c = e/2
And so,
y = (1/2)ln[2(x³ + e/2)]
 = (1/2)ln(2x³ + e)
For the domain,
2x³ + e > 0
2x³ > -e
x³ > -e/2
x > ³√(-e/2)
For the range, y can take all values i.e -∞ < y < ∞

Regards