Need help with these Math-Problems please!?

Question

The first one:
Consider the differential equation dy/dx = 3x²/e^(2y). Find a solution y=f(x) to the differential equation satisfying f(0) = 1/2.

And the second one:
What is the volume of the solid whose base is the region in the xy-plane bounded by the graphs of y=e^(3x), y=e^x, and x=1. if every cross-section cut by a plane perpendicular to the x-axis is a square?

Thank you so much... =) By the way, i really suck at math :)

chauncy · Accepted Answer

Q1) dy/dx = 3x²/e^(2y)

integral e^(2y) dy = integral 3x^2 dx

e(2y)/2 = x^3 + C (where C is a constant)

Require f(0) = 1/2

e^(1/2) = 0 + C

i.e. C = e^(1/2)

e(2y)/2 = x^3 + e^(1/2)

e(2y) = 2[x^3 + e^(1/2)]

2y = ln{2[x^3 + e^(1/2)]}

y = ln{2[x^3 + e^(1/2)]}/2


Q2) integral (x = 0 ---> 1) [e^(3x) - e^x]^2 dx

= integral (x = 0 ---> 1) [e^(6x) - 2e^(4x) + e^(2x)] dx

= [e^(6x)/6 - e^(4x)/2 + e^(2x)/2] | (x = 0 ---> 1)
 
= (e^6/6 - e^4/2 + e^2/2) - (1/6 - 1/2 + 1/2 )

= e^6/6 - e^4/2 + e^2/2 - 1/6

= approx. 43.47

Need help with these Math-Problems please!?

1 Antwort