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    Mathematica Numerical solution in Mathematica

    The main problem is that, Mathematica tries to solve your problem analytically first. So it plugs in symbolic x, and your function can't handle it. Please refer to this question in http://www.voofie.com/concept/Mathematica/" [Broken]...
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    Taylor Series in Multiple Variables

    Your question really depends on the values of x and y. And your expression doesn't have c. I assumed your function to be: f(x,y)=\sqrt{a x^8+b x^4y^4+c y^8} I only outline the method to obtain a power series here. f(x,y)=c y^8\sqrt{1+\left(\frac{a x^8+b x^4y^4}{c y^8}\right)} By...
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    Very interesting problem

    You are welcome. Update us with your problem if you want.
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    Deriving tan power series

    Piano man. Here is a link in http://www.voofie.com/concept/Mathematics/" [Broken] that you maybe interested. http://www.voofie.com/content/117/an-explicit-formula-for-the-euler-zigzag-numbers-updown-numbers-from-power-series/" [Broken] I derived the power series of the function sec x +...
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    Frobenius method questions

    Please substitute y = x^2 to your differential equation, and you will find that it is indeed one of the solution. y = x^2 y' = 2x y'' = 2 Therefore: x^2 y'' -3x y' + 4y = 2 x^2 - 6 x^2 + 4x^2 = 0
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    Another infinite series

    Actually, you can find a close form for your summation expression. Please refer to http://www.voofie.com/concept/Mathematics/" [Broken] for details: http://www.voofie.com/content/156/how-to-sum-sinn-q-and-cosn-q/" [Broken] In short, \sum _{n=-N}^N \cos (n \theta )=\cos (N \theta )+\cot...
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    Euler Equation, Arcsine

    Hello. psholtz. I thought you just do the integration without considering their dependence. It is interesting that x\sqrt{1-y^2} + y\sqrt{1-x^2} = C is a solution. Thank you for your information in the Jacobian as well.
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    Frobenius method questions

    In fact, what you have show is that, x^2 is one of the solution. Since for all other terms other than a_0, they have to be zero in order for the expression to equal to zero.
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    Euler Equation, Arcsine

    I am sorry, but your 2nd equation: \sqrt{1-y^2}dx + \sqrt{1-x^2}dy = 0 cannot be integrated to get the 3rd equation: x\sqrt{1-y^2} + y\sqrt{1-x^2} = C You cannot integrate term by term, from dx to x, since \sqrt{1-y^2} depends on x, and \sqrt{1-x^2} depends on y. In fact, if you...
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    Property Function exp

    I have answered your question. Please have a look. http://www.voofie.com/content/152/how-to-prove-eat-e-at_0-eat-t_0/" [Broken]
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    Inhomogeniouse System

    Hello. Please refer to my article in http://www.voofie.com/concept/Mathematics/" [Broken]: http://www.voofie.com/content/18/solving-system-of-first-order-linear-differential-equations-with-matrix-exponential-method/" [Broken] What you really need is matrix exponential, instead of matrix...
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    Property Function exp

    For your first equation, please refer to this question in http://www.voofie.com/concept/Mathematics/" [Broken]: http://www.voofie.com/content/152/how-to-prove-eat-e-at_0-eat-t_0/" [Broken] I think you typed wrong in this formula: exp(At)_t=0 = I 0 is not equal to I. And your what's your...
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    Two ODE's

    I tried various method in solving the 1st equation, but without any success. Sorry.
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    Two ODE's

    Hello ferry2, I have solved your 2nd equation here: http://www.voofie.com/content/146/how-to-solve-2x12-y--2-2x1-y-4-y-0/" [Broken] And the solution is given by: y(x) = C_1(2x+1) +C_2 (2x+1) \ln (2x+1)
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    Sum of subsets

    Generating Function of your number soroush1358, I have answer your question in this post: http://www.voofie.com/content/144/sum-of-subsets-with-k-elements-having-a-total-sum-of-r/" [Broken]. You can read more Mathematics related articles in http://www.voofie.com/concept/Mathematics/"...
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    Wikipedia shows a proof of product rule using differentials by

    I agree that differential has the meaning of limit already. But without saying so, it is rather difficult to explain why du dv = 0 in PhDorDust's example. Furthermore, I wonder if product of differentials must be zero, since we sometimes have dx dy in double integral. I think the reason is that...
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    AMC Question HARD

    You may just write out the recurrence equation and solve it yourself: a_n = \frac{a_{n+1}+a_{n-1}}{2} +1 a_1=1 a_{12}=12 The solution is: a_n = -n^2 + 14 n -12
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    2nd order homogeneous

    Hello, beetle2. It didn't mentioned 2nd order with variable coefficients, since there is no general method in solving that. However, what you can do is by guessing the first solution. Then you can find the 2nd one using various method, such as...
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    Derivatives wrt functions

    Char. Limit, for the integration, what you have written is the standard way I learnt to do integration. The general form is: \int g(x) f'(x)d x=\int g(x)d f(x) For example, by using the property of differential, d x^2/2 = x d x \int e^{x^2}x \text{dx} =\frac{1}{2}\int...
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    Derive Multivariable Taylor Series

    For a more direct approach, you may try this: f(x+\Delta x, y+\Delta y) = \sum_{k=0}^{\infty} \frac{(\Delta x)^k}{k!} \frac{d^k}{dx^k} f(x,y+\Delta y) = \sum_{k=0}^{\infty} \frac{(\Delta x)^k}{k!} \frac{d^k}{dx^k} \sum_{j=0}^{\infty} \frac{(\Delta y)^j}{j!} \frac{d^j}{dy^j} f(x,y)...
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    Wikipedia shows a proof of product rule using differentials by

    You are right up to this step: d u\cdot v = u \cdot dv + v \cdot du + du \cdot dv However, if differential is small, differential multiplying to another differential is much smaller, which can be neglected (In limit sense). Therefore: d u\cdot v = u \cdot dv + v \cdot du
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    Functional derivative

    Hello, alicexigao. I can't really help you prove the identity. But I think it maybe able to evaluate to something more simple. \int _0^1\frac{d C(q(x)+k'(q'(x)-q(x)))}{d k'}d k' =\int _0^1d C(q(x)+k'(q'(x)-q(x))) =[ C(q(x)+k'(q'(x)-q(x)))]_0^1 =C(q(x)+(q'(x)-q(x)))-C(q(x))...
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    Secant series

    Dickfore is correct about the difficulties in finding inverse of infinity series. I have written a paper about finding power series of tan x + sec x. And I think Unit may take a look in it. For the secant series, it just corresponds to the even power of the series, as tan x is odd, and sec x is...
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    How to differentiate y = 8 ln x - 9 x with respect to x^2?

    quasar987, negative logarithm does indeed exist. http://en.wikipedia.org/wiki/Logarithm#Logarithm_of_a_negative_or_complex_number" But I think you are right to assume x is positive.
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    How to differentiate y = 8 ln x - 9 x with respect to x^2?

    Hello, quasar. I don't think using square root function is a good method, since you don't know whether to take the positive or negative square root. For more difficult case, you may not be able to find the inverse function. To differentiate f(x) w.r.t g(x), just do the following: \frac{d...
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    Double integral of product of Diracs

    The key step that's incorrect should be you replace: F_1=\delta (t-n T)\delta (\tau -n T) with F_2=\delta (t-\tau ) Clearly, F_1 is non-zero when both t = n T and \tau = n T. Though it is true that t = \tau, but you missed t = n T. There is a second problem as well. A product of...
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    Binomial series vs Binomial theorem, scratching my head for three days on this

    I think your original binomial series is wrong too. The correct one should be summing about r, not n. i.e. (1+x)^n = \sum_{r=0}^{\infty} \binom{n}{r} x^r This formula is valid, even for complex n. You are right that this formula can be derived using Taylor series.
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    Very interesting problem

    Your equation can be written as: \frac{\partial ^2(m x)}{\partial t^2}=0 And this can be solved easily. Hope it helps you.
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    Count the solutions in nonnegative integers

    In this article in http://www.voofie.com/concept/Mathematics/" [Broken]: http://www.voofie.com/content/133/repeated-sum-and-partial-difference-equation/" [Broken] I first gave a standard method in solving the problem (probably the same approach as your theorem). Then I solve it using a...
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    Nonhomogeneous Second Order ODE containing log

    If it is not a must that you have to use method of undetermined coefficients, you can have a look of the operator method. In this case, you don't have to care about what kind of non-homogeneous function you have. At least you can write the solution in integral form. Please refer to my tutorial...
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