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1

$$\begin{cases} x'=-6x+2y \\ y'=-20x+6y \end{cases}$$ One can solve it with matrix calculus or with the method of substitution (below) : $2y=x'+6x \quad \to \quad 2y'=x''+6x'$ $$2y'=-40x+12y=-40x+6(x'+6x)=x''+6x'$$ $$x''+4x=0$$ $$x(t)=C_1\cos(2t)+C_2\sin(2t)$$ Then, you can compute $y(t)=\frac{1}{2}x'(t)+3x(t)$


1

As has been pointed out in the other answers, there are four complex numbers $z$ with the property that $z^4=i$. If you would like to understand the behavior exhibited by WolframAlpha, however, you should consider the fact that its square root function is (in fact) a function built on top of Mathematica's Sqrt command. It returns exactly one number, namely ...


0

We can write $i$ as $i=e^{i(\pi/2+2k\pi)}$ for any integer $k$. Then, taking a square root, we see that $$\sqrt{i}=e^{i(\pi/4+k\pi)} \tag 1$$ for any integer $k$. Note that there are two distinct values of $\sqrt i$ in $(1)$; for $k=-1,0$, we have $\sqrt i = e^{-3i\pi/4}$ and $\sqrt i = e^{i\pi/4}, respectively. Taking the square root of the ...


2

Trick is to introduce an unknown, $z$. Write $z=\sqrt{ \sqrt{i}}$. We want to know how many values are possible for $z$. Squaring we get $z^2=\surd i$. Again squaring we get $z^4-i=0$. This is a polynomial of degree 4 over the complex numbers. So it has 4 roots. Taking the derivative shows that the roots are distinct. SO we have 4 distinct values for the ...


8

Your list of four solutions only has two - as has been pointed out you listed two of them twice. (In the original post, anyway...) But there are four roots. First, $i=e^{i\pi/2}=e^{i5\pi/2}$ leads gives two values for $\sqrt i$, namely $e^{i\pi/4}$ and $e^{i5\pi/4}$. Each of these has two square roots: Since $e^{i\pi/4}=e^{i9\pi/4}$ it has the two square ...


2

Yes. Using the angle-addition formula, $$ \sin{(2t-\pi/2)} = \sin{2t}\cos{(\pi/2)}-\cos{2t}\sin{(\pi/2)} = -\cos{2t}. $$


2

Yes, through two identities: $$-4\sin(2t-\pi/2)=-4\sin(-(\pi/2-2t)) \\ =4\sin(\pi/2-2t) \\ =4\cos(2t).$$ The first identity is in the second equation, and is the fact that $\sin$ is odd. The second one is in the third equation, and is the basic cofunction identity (easily seen by drawing a right triangle).


3

Here are some more examples of fractions and the repeating-decimal representations of them in Wolfram Alpha: \begin{align} \frac{1}{1665} = \frac{6}{9990} &= 0.000\overline{600}, \\[.7ex] \frac{617}{49999950} = \frac{1234}{99999900} &= 0.0000\overline{123400}, \end{align} but \begin{align} \frac{67}{666} = \frac{1}{10} + \frac{6}{9990} &= ...



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