Advanced Math Archive: Questions from April 12, 2023
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Determine \( \frac{d y}{d x} \) of the following and simplify if possible: 3.1 \( y=\frac{2 \sin x}{1-\cos x} \) 3.2 \( y=\frac{1}{x}-\cos ^{2}(3 x) \) 3.3 \( y=\sqrt{x} e^{\left(x^{2}+2\right)} \) 3.2 answers -
\( \mathbf{Y}^{\prime \prime}=-\left[\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right] \mathbf{Y}+\left[\begin{array}{c}t \\ t^{2}\end{array}\right], \mathbf{Y}(0)=\left[\begin{array}{l}1 \\ 1\end{arr2 answers -
3. Solve the following ODE using the Laplace Transform: (a) \( y^{\prime \prime}-3 y^{\prime}+2 y=e^{2 t}, y(0)=1, y^{\prime}(0)=0 \) (b) \( y^{\prime \prime}-4 y^{\prime}+5 y=\cos t, y(0)=0, y^{\prim2 answers -
Laplace transform
\( y^{\prime \prime}+2 y^{\prime}+2 y=2 t, \quad y(0)=2, \quad y^{\prime}(0)=-7 \)2 answers -
Solve the previous problem by methods of ordinary differential equations
Una masa que pesa 32 libras alarga 6 pulgadas un resorte. La masa se mueve en un medio que ofrece una fuerza de amortiguamiento que es numéricamente igual a \( \beta \) veces la velocidad instantáne2 answers -
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For \( x, y \in \mathbb{R} \) define \[ d(x, y):=\left|\frac{x}{1+|x|}-\frac{y}{1+|y|}\right| . \] Is \( (\mathbb{R}, d) \) a metric space? Is \( (\mathbb{R}, d) \) complete?2 answers -
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2. Solve the following IVP: \[ y^{\prime \prime}-y=e^{t} \cos (2 t) \quad y(0)=0, y^{\prime}(0)=1 \]2 answers -
Hace dos años Francisco abrió un restaurante turístico en Lurín y como era de esperarse, el plato más solicitado era el chicharrón de chancho. Él abrió su negocio como todo pequeño comerciant2 answers -
Encontrar una solución en serie de potencias en torno a \( \mathrm{x}=0 \) de la ecuación \[ y^{\prime}+2 x y=0 \] Compruebe su solución por alguno de los métodos de solución de EDO de primer ord2 answers -
Sketch the vector fields of the following formulas
\( \boldsymbol{F}[\boldsymbol{X}]=\boldsymbol{F}[x, y]=\left(\begin{array}{l}3 \\ 2\end{array}\right) \) b) \( \boldsymbol{F}[\boldsymbol{X}]=\boldsymbol{F}[x, y]=\left(\begin{array}{c}y / 3 \\ 0\end{0 answers -
(a) Let \[ f(x, y)=\ln \left(x^{2}+y^{2}\right), \quad \forall(x, y) \in \mathbb{R}^{2} \backslash\{(0,0)\} . \] Compute \( \Delta f(x, y) \). (b) Let \[ g(x, y, z)=\frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}},2 answers -
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