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Calculus/Calculus - Separation of Variables

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Expert: - 6/19/2008


Question
Hello, I am doing a problem for Calc. 2 and I can't seem to finish it.

The problem is:

Use separation of varaiables to solve for y explicitly or implicitly in terms of t.

dy/dx = (e^(3y)(sint)^2)/(ysect)

I got all the way up to
-ye^(-3y)/3 - e^(-3y)/9 = -sin^3t/3 + C

And I can't seem to solve for y. Any help would be great!

Answer
Separate the problem into ye^(-3y)dy = sin^2(t)/sec(t)dt.
On the left side do integration by parts and on the right side make the 1/sec be a cos.  This appears to be what you did.
After looking at both sides of your equation for a few minutes, it looks correct.

Facotring the left side gives you -e^(-3y)(y/3 + 1/9).  It would appear all that could be done with this would be to say that
y/3 + 1/9 = -e^(3y)(-sin^3(t)/3) + C).  Note that the two minus signs cancel on the right and C changes sign.  Multiply by 3, giving,
y + 1/3 = 3e^(3y)((sin^3(t)/3 + C).  Moving the 1/3 to the right gives us the answer y = 3e^(3y)((sin^3(t)/3 + C) - 1/3.

If the -1/3 were not there, you could move the exponential y term to the left and integrate again.  Unfortunately, the terms -1/3 is there and that's about as far as we can go.

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