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About Scotto
Expertise
Any kind of mathematics (algebra, geometry, trigonometry, matrices, calculus, linear approximation, linear regression, linear programming, numerical analysis, probability, statistics, etc.). I also have answered some questions in Physics, Chemistry, and Biology. I would like to volunteer in all areas of Mathematics, not just calculus, and the other three courses that were mentioned.
Experience
Experience in the area: I have tutored students in all areas of mathematics for over 20 years.
Education/Credentials: BSand MS in Mathematics from Oregon State University, where I completed sophomore course in Physics and Chemistry. I received both degrees with high honors.
Awards and Honors: I have passed Actuarial tests 100, 110, and 135.
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You are here: Experts > Teens > Homework/Study Tips > Calculus > Calculus - Separation of Variables
Expert: Scotto
Date: 6/19/2008
Subject: Calculus - Separation of Variables
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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