# All Questions

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### Fast way to calculate Eigen of 2x2 matrix using a formula

I found this site: http://www.math.harvard.edu/archive/21b_fall_04/exhibits/2dmatrices/index.html Which shows a very fast and simple way to get Eigen vectors for a 2x2 matrix. While harvard is quite ...
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### Question on a third-order boundary value problems

This is the corollary $2.1$, from the article "Positive solutions of third order semipositone boundary value problems" if $$u'''=\lambda (\sum_{i=1}^{m} c_i(t)u^{\mu_i}-d(t))+e(t), t\in (0,1)$$ and ...
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from Stewart, Precalculus, 5th, p56, Q. 79 Find all real solutions of the equation $$\dfrac{x+5}{x-2}=\dfrac{5}{x+2}+\dfrac{28}{x^2-4}$$ my solution ...
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I have a relationship $y=f(x)$ for which I can obtain data through simulation. I have good reason to suspect that this relationship is quadratic (rather than, say, exponential), and would like to ...
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### Finding the Fourier Series of $\sin(x)^2\cos(x)^3$

I'm currently struggling at calculation the Fourier series of the given function $$\sin(x)^2 \cos(x)^3$$ Given Euler's identity, I thought that using the exponential approach would be the easiest ...
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### Showing it is a joint probability density function

I have two random variables $X,Y$ with a joint density function $f_{X,Y}(x,y)=x+y$ if $(x,y)\in[0,1]\times [0,1]$ and otherwise $f_{X,Y}(x,y)=0$ I want to analyze this case in different cases, first ...
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### calculate kernels of matrices with angles

So my professor gave me this question: I have to find the basis of the eigenvalues of this matrix \begin{pmatrix} \cos(q) & \sin(q)\\ \sin(q) & -\cos(q)\\ \end{pmatrix} so I calculate ...
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### If $x \leq g(x) \leq x^2-x+1$ where $x \in [0,2]$, can we say that $g(x)$ is continuous at $x=1$?

If $x \leq g(x) \leq x^2-x+1$ where $x \in [0,2]$, can we say that $g(x)$ continuous at $x=1$ ? Is $g(x)$ continuous in $[0,2]$?
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### For every prime of the form $2^{4n}+1$, 7 is a primitive root.

What I want to show is the following statement. For every prime of the form $2^{4n}+1$, 7 is a primitive root. What I get is that $$7^{2^{k}}\equiv1\pmod{p}$$ ...
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### For what $p$ is $x^p$ Lebesgue Integrable?

Revising for an exam on Monday any help with the following question would be greatly appreciated; If $f$ is a function on $(0, \infty)$ taking values in $\mathbb R$, defined $f(x)=x^p$ ($p$ is a real ...