Homework 8 collaboration area
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From Mnestero:
So after a bunch of algebra to solve the system of equations on prob 12 of 6.7 I got an answer. I often make simple mistakes, so I wanted to see if anyone else got what I have:
y1 = cos(sqrt(2)t)+ 2/5 cos(t)- 7/5 cos(sqrt(6)t) y2 = 1/5 cos(t) + 14/5 cos(sqrt(6)t)
From Eun Young:
If you hit the system of differential equations by the Laplace transform, you'll get
$ S^2Y_1 - S = -2 Y_1 + 2Y_2 $ and $ S^2Y_2 - 3S = 2Y_1 - 5Y_2 $. This is a system of two equations. Solve this for $ Y_1 $ and $ Y_2 $ using Cramer's rule or just algebra. Then,
$ Y_1 = \frac{s^3+11s}{s^4+7s^2+6} $. Find $ y_1 $ using partial fractions. It's similar for $ y_2 $
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From Chris:
The example in the book and in our notes doesn't look the same as the problem 5 in 12.12. I'm not even sure how to set up the problem. Can anyone help get me started?
From Mnestero:
I started this problem by taking the laplace with respect to t. This gave me s*W = x/s^2 - x d/dx * W. I took the derivative of x to be 1. I then solved for W, which gave me W = x/(s^2(s+1)). After this use partial fractions. I am not positive that this is the correct approach - but it matches the answer in the back of the book. Anyone else have any thoughts?
From Eun Young:
If you take the Laplace transform with respect to t you'll have
$ x \frac{\partial W}{\partial x} + s W = \frac{1}{s^2} $.
Divide both sides by x then you'll have
$ \frac{\partial W}{\partial x} +\frac{ s}{x} W = \frac{x}{s^2} $.
This is a first-order linear ODE. Be careful. s is a constant. See section 1.5 for reference.
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On prob 16 of 11.1, I set up three piece wise functions to find Bn (odd function An and Ao are 0). From -pi to -pi/2 I set F(x)=0 From -pi/2 to pi/2 I set F(x)=x From pi/2 to pi I set F(x)=0
From here I solved for Bn. The fourier series I calculated is F(x) = 2/pi sinx - 2/4pi sin2x + 2/9pi sin3x ... When graphing this, it is similar to the original graph, but seems slightly off. Am I setting up the problem wrong?
From Andrew:
I don't believe we have to do prob 16 of section 11.1 for the homework, only 11.1.12, 11.1.14, and 11.1.18
From Michael:
Yeah, you're right. Well I guess I got some extra practice in.
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I suppose there is a more elegant way to get through 6.6 #8 that direct integration...is this valid? f(t) = sint Use shifting theorem, so from the table, (1+k)/(s^2 + (1+k)^2) and by differentiation of transforms, L{tf(t)} = - derivative( (1+k)/(s^2 + (1+k)^2) ) ?
Thanks!
-Christine
