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    Simple Integral

    Nevermind. I guess the second part of the problem is incorrect. Mathematica has arrived at the same conclusion as this also. Thanks for all the help.
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    Simple Integral

    Thank you Dick and Jgens. \int{x^3}\sqrt{1+x^2}dx Let u=1+x^2\rightarrowdu=2xdx \int{x^3}\sqrt{1+x^2}dx=\frac{1}{2}\int(u-1)\sqrt{u}du \frac{1}{2}\int(u-1)\sqrt{u}du=\frac{1}{2}(\int{u^{\frac{3}{2}}}du-\int{u^{\frac{1}{2}}du) \int{u^{\frac{3}{2}}}du=\frac{2}{5}u^{\frac{5}{2}}...
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    Simple Integral

    Homework Statement Hello. I have a simple integral here that has been stumping me for the last 30 minutes. It appears that my basic integration skills have gotten very rusty. Homework Equations \int{x^3}\sqrt{1+x^2}dx The Attempt at a Solution I am pretty sure a simple...
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    How fast is the distance changing? (FODE, I think)

    Thanks. This makes sense to me also. I would like to point out something about this problem. I am not sure how relevant this is because the "correct" answer could have come from anywhere. I found this problem on another forum and could not solve it. The original poster came up with the same...
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    How fast is the distance changing? (FODE, I think)

    Hmm, ok. Using your suggestions here is what I did: I re-wrote my equations (Thanks, organization is crucial) as: C(t)=\sqrt{(50t)^2+(27t)^2} I then simplified as follows: C(t)=\sqrt{t^2(50^2+27^2)} ...then: C(t)=\sqrt{t^2(3229)} ...and finally: C(t)=t{\sqrt{3229}}...
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    How fast is the distance changing? (FODE, I think)

    Sorry about the abbreviation. In my DiffEq book, they use FODE to mean First Order ordinary Differential Equation. This problem does not come from my DiffEq book, but it did remind me of a similar problem in that book that also stumped me. Thank you for the hint. As always your posts...
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    How fast is the distance changing? (FODE, I think)

    Homework Statement 2 cars start from the same point. Car A travels a constant 50 mph due west. Car B travels a constant 27 mph due south. After 3 hours, how fast is the distance changing between them? Homework Equations The Attempt at a Solution I saw this problem online...
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    Laplace Eq with Dirichlet boundary conditions in 2D (solution check)

    First, I forgot to thank you for taking the time to reply. Thank you. Gosh, I see. Sorry about the earlier post. I was thinking of only including the even numbers for some reason. Would you say that what you have posted above is "as good as it gets"? I was concerned about how I...
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    Laplace Eq with Dirichlet boundary conditions in 2D (solution check)

    Are you suggesting an index change? As in swapping out all n's to the right of sigma with m, where m = 2n? Or changing the n's to n(n+1)?
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    Laplace Eq with Dirichlet boundary conditions in 2D (solution check)

    Homework Statement The steady state temperature distribution, T(x,y), in a flat metal sheet obeys the partial differential equation: \frac{\partial^2{T}}{\partial{x}^2}+{\frac{\partial^2{T}}{\partial{y}^2}}=0 Separate the variables and find T everywhere on a square flat plate of sides S with...
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    How to check Fourier series solution (complex)

    Thanks a lot! That is also the answer I got. I realized why maple wouldn't plot it, it was because I did not account for the "n" in the denominator. No wonder Maple was blabbering about a singularity.
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    How to check Fourier series solution (complex)

    Thanks for the clarification, Vela. I spoke with my professor today, and he also said the original integral should be correct. He went on to say that I should be able to plot it in Maple. So, I guess that means I need some more practice in Maple (That should be no surprise to...
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    How to check Fourier series solution (complex)

    Thanks for pointing that out. I'll recalculate my coefficient value later today. My tutorial lists the equation for c_n as: c_n=\frac{1}{\lambda}\int_{x_0}^{x_0+\lambda}e^{-i{k_n}x}f(x)dx In a paragraph above this equation, it states that: "...But most applications involve either...
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    How to check Fourier series solution (complex)

    I found the coefficients, c_n, by integrating: c_n=\int_{0}^{1}t(1-t)e^{-i2\pi{n}t}dt Are you saying that this is not the correct method to find c_n?
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    Diff eq.

    Wouldn't that be: x(t)=\frac{(tan(\frac{t-14.012}{9}))}{9}
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    How to check Fourier series solution (complex)

    Homework Statement Find the complex Fourier series for: f(t)=t(1-t), 0<t<1 Homework Equations f(t)=\sum_{n=-\infty}^{\infty}c_n{e^{i\omega_n{t}}} c_n=\frac{1}{\tau}\int_{t_0}^{t_0+\tau}e^{-i\omega_n{t}}f(t)dt \omega_n=2\pi{n}\quad\tau=1 The Attempt at a Solution I solved...
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    Writing a polynomial in terms of other polynomials (Hermite, Legendre, Laguerre)

    Wow. Thanks. I must have made a mistake along the line. I haven't found it yet, but I agree with your post. Thank you for doubling checking and taking the time to evaluate the integral. Thanks again, I seem to make a ton of algebraic / input mistakes.
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    Writing a polynomial in terms of other polynomials (Hermite, Legendre, Laguerre)

    Thanks for the response. If, a_n=\int_{0}^{\infty}f(x)\phi_n(x)e^{-x}dx For the laguerre polynomials is correct, then does this say: a_4=\int_{0}^{\infty}(2x^4-x^3+3x^2+5x+2)(\frac{1}{24}x^4-\frac{2}{3}x^3+3x^2-4x+1)e^{-x}dx I tried the above equation. I ended up having 9 integrals that...
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    Writing a polynomial in terms of other polynomials (Hermite, Legendre, Laguerre)

    Homework Statement The first 3 parts of this 4 part problem were to derive the first 5 Hermite polynomials (thanks vela), The first 5 Legendre polynomials, and the first 5 Laguerre polynomials. Here is the last part: Write the polynomial 2x^4-x^3+3x^2+5x+2 in terms of each of the sets of...
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    Indefinite integral (Hermite polynomials)

    Now it seems that I either have the wrong solution, or I do not know what I am doing. When I use the relation : \int_0^\infty{x^m}x^n{e^{-x^2}dx=\frac{1}{2}((m+n)-1/2)! if (m+n) = even I'm assuming the gamma function below is equal to the right hand side above...
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    Indefinite integral (Hermite polynomials)

    Thanks for catching my mistake! So after scouring the internet (I wasn't able to figure out the quantity by searching or using the 2 bits of information) and putting together all I found, I think: \Gamma(n+\frac{1}{2})=\frac{2n!}{n!2^{2n}}\sqrt\pi I realize the above quantity needs to be...
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    Indefinite integral (Hermite polynomials)

    Thanks vela! I am not sure how the gamma function works. I have just looked over articles pertaining to it online, but I am not confident enough in my understanding of it to be able to identify it in this case. Here is what I have following vela's advice...
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    Indefinite integral (Hermite polynomials)

    Homework Statement I need to evaluate the following integral: \int_{-\infty}^{\infty}x^mx^ne^{-x^2}dx I need the result to construct the first 5 Hermite polynomials. Homework Equations The Attempt at a Solution First I tried arbitrary values for "m" and "n". I was not able to...
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    Fourier series proof

    Thanks! I can't believe I burned almost 3 hours on that, only to have the answer as soon as I posted it.
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    Fourier series proof

    Something just dawned on me. Does anyone think the following is correct?: If f(x) is symmetric, then b_n is 0 since sine is antisymmetric. Therefore, the sin term in the Fourier series drops out. If f(x) is antisymmetric, then a_n is 0 since f(x) is antisymmetric and the cosine term in the...
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    Fourier series proof

    Homework Statement Suppose, in turn, that the periodic function is symmetric or antisymmetric about the point x=a. Show that the Fourier series contains, respectively, only cos(k_{n}(x-a)) (including the a_0) or sin(k_{n}(x-a)) terms. Homework Equations The Fourier expansion for the...
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    Orthogonality integral help

    Thank you for reading and verifying. Based off this I believe that completeness means that any vector in R^3 can be expressed/written as a linear combination of the 3 basis vectors.
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    Orthogonality integral help

    Thank you for the reply This makes sense to me. If the phi's are an infinite set of functions, does that mean in my problem it represents every possible vector in 3-D? If that is the case, why does the problem word it as "the completeness of a set of basis vectors..."? I read the set of...
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    Orthogonality integral help

    Homework Statement If there exists no function, f(x), except zero, with the property that \int_{a}^{b}{\phi_{n}(x)}f(x)w(x)dx=0 for all \phi_{n}, then the set {\phi_{n}(x)} is said to be complete. Write a similar statement expressing the completeness of a set of basis vectors in...
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