Advanced Math Archive: Questions from September 01, 2022
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Ejercicio 2.1. Considere el grupo \( (A(\mathbb{Z}), \circ) \) (ver (3)) y sean \( f, g \in A(\mathbb{Z}) \) dadas por: \[ \begin{aligned} f: \mathbb{Z} & \longrightarrow \mathbb{Z} \\ x & \longrighta1 answer -
\[ y=A \sin (\omega \mathrm{t}+\gamma)=A(\sin \omega \mathrm{t} \cos \gamma+\sin \gamma \cos \omega \mathrm{t}) \] If gamma is small, what happens? \[ y=A \sin \omega t+A \gamma \cos \omega t \]1 answer -
\[ y=A \sin (\omega \mathrm{t}+\gamma)=A(\sin \omega \mathrm{t} \cos \gamma+\sin \gamma \cos \omega \mathrm{t}) \] If gamma is small, what happens? \[ y=A \sin \omega t+A \gamma \cos \omega t \]1 answer -
only do number 3
denote derivatives with respect to \( x \). 1. \( y^{\prime}=3 x^{2} ; y=x^{3}+7 \) 2. \( y^{\prime}+2 y=0 ; y=3 e^{-2 x} \) 3. \( y^{\prime \prime}+4 y=0 ; y_{1}=\cos 2 x, y_{2}=\sin 2 x \)1 answer -
(b) For \( x, y \in \mathbb{R} \) define \( \rho(x, y)=\sqrt{|x-y|} \). Is \( \rho \) a metric on \( \mathbb{R} \) ?1 answer -
Graph the function. \[ h(x)=\left\{\begin{array}{ll} 2 x & \text { if } 0 \leq x \leq 5 \\ x+5 & \text { if } 530 \end{array}\right. \]1 answer -
1 answer
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1 answer
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Just (H) please and thank you.
Verify that the function is a solution of the differential equation on some interval, for any choice of the arbitrary constants appearing in the function. (a) \( y=c e^{2 x} ; \quad y^{\prime}=2 y \)1 answer -
1 answer
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4A,4B,4C
Solve the initial value problem. (a) \( y^{\prime}=-x e^{x}, \quad y(0)=1 \) (b) \( y^{\prime}=x \sin x^{2}, \quad y\left(\sqrt{\frac{\pi}{2}}\right)=1 \) (c) \( y^{\prime}=\tan x, \quad y(\pi / 4)=31 answer -
3A,3B,3C
3. Find all of the equation. (a) \( y^{\prime}=-x \) (b) \( y^{\prime}=-x \sin x \) (c) \( y^{\prime}=x \ln x \) (d) \( y^{\prime \prime}=x \cos x \) (e) \( y^{\prime \prime}=2 x e^{x} \) (f) \( y^{\p1 answer -
2A -2H
Verify that the function is a solution of the differential equation on some interval, for any choice of the arbitrary constants appearing in the function. (a) \( y=c e^{2 x} ; \quad y^{\prime}=2 y \)1 answer -
Determine and represent geometrically the domain of each of the following functions. b) \( f(x, y)=2+\sqrt{6-y^{2}} \). c) \( h(x, y)=\sqrt{x}+\sqrt{y}+\sqrt{z} \). d) \( h(x, y)=\sqrt{x y} \). e) \(1 answer -
\( y^{\prime \prime}+y=\tan x ; \quad y=-(\cos x) \ln (\sec x+\tan x) \) \( (y-x) y^{\prime}=y-x+8 ; \quad y=x+4 \sqrt{x+2} \) \( 2 x y d x+\left(x^{2}-y\right) d y=0 ; \quad-2 x^{2} y+y^{2}=1 \)1 answer