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##### Test 2 / Re: 2022 midterm 2 solution: Possible typos?

« Last post by**Victor Ivrii**on

*March 25, 2023, 06:48:46 AM*»

Thanks! Fixed

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Thanks! Fixed

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Section 4.1ODE is not enough. You need to take into account also boundary conditions. See Section 4.1

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I've been looking through the midterm solutions from 2022 to study for the midterm coming up. I was wondering if it's possible there are typos in the solutions, or if you can explain the following if I am misunderstanding something:

1. In problem 1 of Main Sitting 2022, there is the equation T' + 4λT = 0. I believe this should be T'' + 4λT = 0, with the first T as a second derivative.

2. In problem 1 of Morning Sitting 2022, there is equation 4T'' + λT = 0. I believe this should be T'' + 9λT = 0.

In the same question, T_{n} = A_{n} cos(nt) + B_{n} sin(nt). I believe this should be T_{n} = A_{n} cos(3nt) + B_{n} sin(3nt).

In the same question, there are initial boundary conditions dependent on the value of t. I believe in the solution when plugging these in, they are swapped.

3. In problem 3 of Main Sitting 2022, there u(r,θ) =B_{0} /2 + (a sum). Then u(1,θ) = A_{0} /2 + (a sum). I'm not sure how B_{0} turned into A_{0} in this.

Thank you!

1. In problem 1 of Main Sitting 2022, there is the equation T' + 4λT = 0. I believe this should be T'' + 4λT = 0, with the first T as a second derivative.

2. In problem 1 of Morning Sitting 2022, there is equation 4T'' + λT = 0. I believe this should be T'' + 9λT = 0.

In the same question, T

In the same question, there are initial boundary conditions dependent on the value of t. I believe in the solution when plugging these in, they are swapped.

3. In problem 3 of Main Sitting 2022, there u(r,θ) =B

Thank you!

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Hi Prof:

I have a question for solving the Laplacian operation using the separation of variables.

Slide: Week 9 Lecture 1 Page 4.

The queston happens in solving ${\theta}'' + \lambda \theta = 0$.

Q: I do not understand how do you derive the $\lambda_{n} = n^2$.

Also, I do not understand why we can get a expression for $\lambda$ ? To my understading, $\lambda$ is just a constant in the ODE.

Thanks

I have a question for solving the Laplacian operation using the separation of variables.

Slide: Week 9 Lecture 1 Page 4.

The queston happens in solving ${\theta}'' + \lambda \theta = 0$.

Q: I do not understand how do you derive the $\lambda_{n} = n^2$.

Also, I do not understand why we can get a expression for $\lambda$ ? To my understading, $\lambda$ is just a constant in the ODE.

Thanks

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If domain is "simple" in the appropriate coordinate system it is a good idea to try to find such solutions. For linear equations we even can construct general solutions as linear combinations of such solutions. Details in the class later.

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When is it a good idea to assume solutions are separable? Are there any implied assumptions about the nature of the solutions when we assume that they are separable?

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I will post solutions to show that all the integrals can be computed using elementary functions and $\operatorname{erf}(.)$

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Dear Prof Ivrii,

I did some home assignment for heat equation problems to prepare the quiz, and I found some integrals seems hard to compute. So I am wondering if we need to compute the integral completely correct to get the full mark when do the quiz, or just simplify the heat formula is fine? Thank you.

I did some home assignment for heat equation problems to prepare the quiz, and I found some integrals seems hard to compute. So I am wondering if we need to compute the integral completely correct to get the full mark when do the quiz, or just simplify the heat formula is fine? Thank you.

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Yes––as long as you post your variant solution

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Just a quick question, will the quiz solution be posted every week? Thanks.