Advanced Math Archive: Questions from March 03, 2023
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2 answers
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Solve the following differential equations. \[ \frac{d y}{d t}-\frac{y}{t}=13 t e^{t} \] \[ \begin{array}{l} y=13 t e^{t}+C t \\ y=13\left(t e^{t}-2 e^{t}-\frac{2}{t} e^{t}+C t^{-1}\right) \\ y=13 t e2 answers -
use the sándwich theorem to find the limit
3.-Use el teorema del sandwich para encontrar el límite de \[ \lim _{n \rightarrow \infty} \frac{1}{\sqrt{n^{8}+n^{16}}} \]2 answers -
Find \( f^{\prime}(x) \) or \( d y / d x \). (Do not simplify the answers, show each step.) 7. \( y=\left(4 x^{2}+1\right)^{2} /(2 x-3)^{3} \) 8. \( f(x)=\tan ^{2}(3 x+4) \) 9. \( y=\sqrt{\sqrt[3]{4 x2 answers -
19. Sea \( V=\mathcal{M}_{n n} \) el espacio de las matrices de orden \( n \times n \). a) Definamos \( \langle A, B\rangle=\operatorname{tr}\left(A B^{T}\right) \), donde \( \operatorname{tr} A=\sum_2 answers -
25. Sea \( M \) un subespacio de un espacio de Hilbert \( V \). Entonces a) \( M \) es completo si y sólo si \( M \) es cerrado en \( V \). b) \( M^{\perp}=\{0\} \) si y sólo si \( M \) es denso en2 answers -
5. Probar que la función \( \rho: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \) definida por \[ \rho(x, y):=|y-x|^{1 / 2} \] es una distancia en \( \mathbb{R} \).2 answers -
6. Sean \( X \) un espacio normado, \( Y \) un espacio vectorial y \( f: X \rightarrow Y \) una aplicación lineal e invectiva. Probar que, definiendo \[ \|y\|=\|f(y)\| \quad \forall y \in Y \] se obt2 answers -
Sea \( V=C([a, b]) \) a) Pruebe que \( V \) es un espacio de Banach con la norma \( \|f\|_{\infty}=\max _{a \leq t \leq b}|f(t)| \) para todo \( f \in V \) y para todo \( t \in[a, b] \). b) Pero con l2 answers -
Solve the following partial differential equations: \[ \begin{array}{l} \text { a) } \nabla^{2} / u_{4}=\partial^{2} u_{4} / \partial x^{2}+\partial^{2} u_{4} \partial y \\ u 4(0, y)=g 1(y) \quad u 4(2 answers -
2 answers
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Express the given function \( f \) in the form \( f(z)=u(x, y)+i v(x, y) \). \[ f(z)=e^{9 \bar{z}+i} \] \[ u(x, y)= \] \[ v(x, y)= \]2 answers -
2. Use the method of Laplace transforms to solve the following IVP a. \( y^{\prime \prime}-y^{\prime}-6 y=0 ; \quad y(0)=3, \quad y^{\prime}(0)=4 \) b. \( y^{\prime \prime}-2 y^{\prime}+5 y=0 ; \quad2 answers -
(b) \( [\mathbf{1} \mathbf{p t}] \) Compute \[ \iint_{D}\left|\frac{x+y}{\sqrt{2}}-x^{2}-y^{2}\right| d A, \quad D=\left\{(x, y) \mid x^{2}+y^{2} \leq 1\right\} \]2 answers -
Find \( y^{\prime \prime} \) if implicit differentiation produces the following equation in \( y^{\prime} \) : \[ y^{\prime} \cdot \tan \left(\sqrt[3]{x^{2}+5}\right)=\csc ^{5}\left(y^{3}\right)-e^{52 answers -
1. Define \( A_{\alpha}=\left\{(x, y) \in \mathbb{R}^{2} \mid 0 \leq x \leq \alpha, 0 \leq y \leq \frac{-(1-\alpha)}{\alpha} x+(1-\alpha)\right\} \), where \( 02 answers -
Solve the following differential equation:
\( y^{\prime \prime}-4 y^{\prime}+13 y=5 e^{2 x}+6 \sin 2 x \)2 answers -
( 1 point) Find \( y \) as a function of \( t \) if \[ 144 y^{\prime \prime}+312 y^{\prime}+169 y=0 \] \[ y(0)=6, \quad y^{\prime}(0)=9 \]2 answers -
(1 point) Find \( y \) as a function of \( t \) if \[ 1296 y^{\prime \prime}+504 y^{\prime}+49 y=0 \] \[ y(0)=5, \quad u^{\prime}(0)=5 \text {. } \] \[ y= \]2 answers -
Find the solution of \[ y^{\prime \prime}+4 y^{\prime}+4 y=448 e^{6 t} \] with \( y(0)=4 \) and \( y^{\prime}(0)=1 \). \[ y= \]2 answers -
please show steps
(1 point) Find \( y \) as a function of \( t \) if \[ 4 y^{\prime \prime}+4 y^{\prime}+y=0 \] \[ y(0)=4, \quad y^{\prime}(0)=7 \]2 answers -
Los polinomios de Taylor de la funcion f(x) = 24x + 8/13 + 9x^2 de grado dos y tres alrededor de cero son у P2 (x) = 0.888889 + 2.66667x + (-0.790125)(x^2) y P3 (x)= 0.888889 + 2.66667x + (-0.790125)
DefiniciondeerroresAV: Problem 4 (1 point) (1 point) Los polinomios de Taylor de la funcion \( f(x)=\frac{24 x+8}{9+8 x^{2}} \) de grado dos y tres alrededor de cero son (1 point) Los polinomios de T2 answers -
iii. \( y^{\prime \prime}+2 y^{\prime}-3 y=u_{2}(t) \sin (t-2), \quad y(0)=0, y^{\prime}(0)=3 \) iv. \( y^{\prime \prime}+2 y^{\prime}+2 y=10 \sin t, \quad y(0)=0, \quad y^{\prime}(0)=1 \). v. \( y^{\2 answers