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
S2Y1 − S = − 2Y1 + 2Y2 and S2Y2 − 3S = 2Y1 − 5Y2. This is a system of two equations. Solve this for Y1 and Y2 using Cramer's rule or just algebra. Then,
$ Y_1 = \frac{s^3+11s}{s^4+7s^2+6} $. Find y1 using partial fractions. It's similar for y2
I'm trying to complete the partial fractions for Y1 and am getting stuck at "B = -sqrt(6) i (1-A)." Any thoughts? I'm not able to break down the values of A and B (or C and D) unless there is another step I've missed. Thanks. -Christine
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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) ) ?
Also wondering about #16. I have done a bit of algebra to get to the form (2s+6)/ (s^2 (s+6)^2 + 20 (s+5)(s+1))
My CAS tells me the answer is e^-3t t sin t but I don't see it in there.
Thanks!
-Christine
From Ryan Leemhuis:
Hi Christine,
The way I approached this problem was to recognize that we have a transform identity for e^-(kt)*sin(t) in the identity table of 6.1. Using the differentiation theorem of laplace transforms we know that multiplying the function by t acts as the negative differentiation of the laplace transform. All you should have to do is take the derivative of the identity given in the table and multiply by -1. ---
Question from Luo Shibo
for problem 12.12, I convert the problem to the ODE below;
$ \frac{\partial W}{\partial x} +\frac{ s}{x} W = \frac{x}{s^2} $.
I treat s as a constant, and x as variable.
Since there is $ \frac{ s}{x} W $,I find it's difficult for me to solve this ODE, can any one give me some help?
