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==Homework 8 collaboration area==
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== Homework 8 collaboration area ==
  
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From Mnestero:
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From Mnestero:  
  
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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:
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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:  
  
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y1 = cos(sqrt(2)t)+ 2/5 cos(t)- 7/5 cos(sqrt(6)t)
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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)  
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y2 = 1/5 cos(t) + 14/5 cos(sqrt(6)t)
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<br> From [[User:Park296|Eun Young]]:
  
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From [[User:Park296|Eun Young]]:
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If you hit the system of differential equations by the Laplace transform, you'll get
  
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If you hit the system of differential equations by the Laplace transform, you'll get
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<span class="texhtml">''S''<sup>2</sup>''Y''<sub>1</sub> − ''S'' =  − 2''Y''<sub>1</sub> + 2''Y''<sub>2</sub></span> and <span class="texhtml">''S''<sup>2</sup>''Y''<sub>2</sub> − 3''S'' = 2''Y''<sub>1</sub> − 5''Y''<sub>2</sub></span>. This is a system of two equations. Solve this for <span class="texhtml">''Y''<sub>1</sub></span> and <span class="texhtml">''Y''<sub>2</sub></span> using Cramer's rule or just algebra. Then,
  
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<math>S^2Y_1 - S = -2 Y_1 + 2Y_2</math> and <math>S^2Y_2 - 3S = 2Y_1 - 5Y_2</math>.
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<math>Y_1 = \frac{s^3+11s}{s^4+7s^2+6}</math>. Find <span class="texhtml">''y''<sub>1</sub></span> using partial fractions. It's similar for <span class="texhtml">''y''<sub>2</sub></span>  
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This is a system of two equations. Solve this for <math>Y_1 </math> and <math>Y_2</math> using Cramer's rule or just algebra.
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Then,
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<math>Y_1 = \frac{s^3+11s}{s^4+7s^2+6}</math>.
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---
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Find <math>y_1</math> using partial fractions. It's similar for <math>y_2</math>
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From Chris:
  
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From Chris:
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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?
  
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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?
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From Mnestero:
  
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From Mnestero:
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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?
  
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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?
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<br> From [[User:Park296|Eun Young]]:  
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From [[User:Park296|Eun Young]]:
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If you take the Laplace transform with respect to t you'll have  
 
If you take the Laplace transform with respect to t you'll have  
  
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<math>x \frac{\partial W}{\partial x} + s W = \frac{1}{s^2}</math>.
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<math>x \frac{\partial W}{\partial x} + s W = \frac{1}{s^2}</math>.  
  
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Divide both sides by x then you'll have
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Divide both sides by x then you'll have  
  
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<math> \frac{\partial W}{\partial x} +\frac{ s}{x} W = \frac{x}{s^2}</math>.
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<math> \frac{\partial W}{\partial x} +\frac{ s}{x} W = \frac{x}{s^2}</math>.  
  
 
This is a first-order linear ODE. Be careful. s is a constant. See section 1.5 for reference.  
 
This is a first-order linear ODE. Be careful. s is a constant. See section 1.5 for reference.  
  
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---
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---  
  
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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).  
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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  
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From -pi to -pi/2 I set F(x)=0
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From -pi/2 to pi/2 I set F(x)=x
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From pi/2 to pi I set F(x)=0
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From here I solved for Bn. The fourier series I calculated is F(x) = 2/pi sinx - 2/4pi sin2x + 2/9pi sin3x ...
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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?  
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When graphing this, it is similar to the original graph, but seems slightly off. Am I setting up the problem wrong?
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From Andrew:
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From Andrew:  
  
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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
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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  
  
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From Michael:
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From Michael:  
  
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Yeah, you're right. Well I guess I got some extra practice in.
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Yeah, you're right. Well I guess I got some extra practice in.  
  
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<br> ---
  
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---
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I suppose there is a more elegant way to get through 6.6 #8 that direct integration...is this valid?
  
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I suppose there is a more elegant way to get through 6.6 #8 that direct integration...is this valid?
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f(t) = sint
  
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f(t) = sint
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Use shifting theorem, so from the table, (1+k)/(s^2 + (1+k)^2)  
  
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Use shifting theorem, so from the table, (1+k)/(s^2 + (1+k)^2)
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and by differentiation of transforms, L{tf(t)} = - derivative( (1+k)/(s^2 + (1+k)^2) )&nbsp;?
  
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and by differentiation of transforms, L{tf(t)} = - derivative( (1+k)/(s^2 + (1+k)^2) ) ?
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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))  
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Also wondering about #16. I have done a bit of algebra to get to the form
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(2s+6)/ (s^2 (s+6)^2 + 20 (s+5)(s+1))
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My CAS tells me the answer is e^-3t t sin t but I don't see it in there.  
 
My CAS tells me the answer is e^-3t t sin t but I don't see it in there.  
  
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Thanks!
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Thanks!  
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                     -Christine
 
                     -Christine
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---
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Question from Luo Shibo
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for problem 12.12, I convert the problem to the ODE below;
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<math> \frac{\partial W}{\partial x} +\frac{ s}{x} W = \frac{x}{s^2}</math>.
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I treat s as a constant, and x as variable
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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?
 +
  
 
[[2013 Fall MA 527 Bell|Back to MA527, Fall 2013]]  
 
[[2013 Fall MA 527 Bell|Back to MA527, Fall 2013]]  
  
 
[[Category:MA527Fall2013Bell]] [[Category:MA527]] [[Category:Math]] [[Category:Homework]]
 
[[Category:MA527Fall2013Bell]] [[Category:MA527]] [[Category:Math]] [[Category:Homework]]

Revision as of 17:48, 21 October 2013

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

---

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.

---

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.


---

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

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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?


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