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Advanced Math Archive: Questions from January 25, 2023

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  2. Determine the inverse of the function \( m(x)=5(-x+3) \). a. \( y=-\frac{x}{5}-3 \) c. \( y=-3(x-5) \) b. \( y=-\frac{x}{5}+3 \) d. \( y=-\frac{x-3}{5} \)
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• Solve 21 🙏🏻
  In Problems 19-21, solve the given initial value problem. 19. \( y^{\prime \prime \prime}-y^{\prime \prime}-4 y^{\prime}+4 y=0 \); \[ y(0)=-4, \quad y^{\prime}(0)=-1, \quad y^{\prime \prime}(0)=-19 \]
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  Question 2. Compute the gradient of the following functions 1. \( f(x, y)=x^{2} y^{2} \) 2. \( f(x, y)=x^{2}+y^{2} \) 3. \( f(x, y)=\mathrm{e}^{x^{2}}+x \ln (y) \) 4. \( f(x, y, z)=z^{4} y+3 x z^{2}+x
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  \[ \begin{array}{l} \mathrm{CV}=\gamma+\alpha \log t \\ \mathrm{ST}=\frac{\rho}{t}+v \end{array} \] Find the values of \( \gamma, \alpha, \rho \) and \( v \) using regression.
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  Solve the differential equation \[ \ln y \frac{d y}{d t}-t y=0 \] \[ \begin{array}{l} \frac{\ln ^{2} y}{2}=\frac{t^{2}}{2}+c \ln t \\ \frac{\operatorname{Ln}^{2} y}{2}=\frac{t^{2}}{2}+c \\ O \frac{\ln
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  1. Given the continuous signal \( f(x, y)=\cos (2 \pi(x-y)) \), evaluate the following: (2 marks each) (a) \( f(x, 1) \delta(x-2, y) \) (b) \( f(x, y) * \delta(x-2, y+1) \) (c) \( \int_{-\infty}^{\inf
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• Solve this Bernoulli DE: dy/dx = x(y^4) - y. y > 0
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• Solve this Bernoulli DE: dy/dx = x(y^4) - y. y > 0
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• Exact Differential Equations
  \( (\sin y-\sin x) d x+(1+x) \cos y d y=0 \quad y(0)=\pi \)
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• use Laplace transform
  \( y^{\prime}+2 y=e^{-t} ; y(0)=1 \) \( y^{\prime}-2 y=1-t ; y(0)=4 \) \( y^{\prime \prime}+y=1 ; y(0)=6, y^{\prime}(0)=0 \)
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• use Laplace transform
  \( y^{\prime \prime}-4 y^{\prime}+4 y=\cos (t) ; y(0)=1, y^{\prime}(0)=-1 \) \( y^{\prime \prime}+9 y=t^{2} ; y(0)=y^{\prime}(0)=0 \) \( y^{\prime \prime}+16 y=1+t ; y(0)=-2, y^{\prime}(0)=1 \)
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  Find \( \frac{d y}{d x} \) : a. \( y=6 x^{3}+9 x+17 \) b. \( y=\ln 4 x^{4} \) c. \( y=g(x)+f(x)+h(x) k(x) \) d. \( y=\frac{4 x^{5}+8 x^{4}}{2 x^{3}} \) Find \( d y \) : Note: subscripts indicate a dif
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• use Laplace transform
  9. \( y^{\prime \prime}+16 y=1+t ; y(0)=-2, y^{\prime}(0)=1 \) 10. \( y^{\prime \prime}-5 y^{\prime}+6 y=e^{-t} ; y(0)=0, y^{\prime}(0)=2 \)
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  Cual opción es la solución a la ED, con condiciones iniciales \( x=3, y=\ln (81) \), siguiente: \[ -y+x \frac{d y}{d x} \ln (x)=0 \] \[ \begin{array}{l} y=4 \ln (x) \\ y=C \ln (x) \\ y=-\ln (3)+\ln
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  Cual opción es la solución a la ED, con condiciones iniciales \( x=10, y=0 \), siguiente: \[ e^{y^{2}} y \frac{d y}{d x}=-5+x \] \[ y^{2}=\ln \left(C\left(-10 x+x^{2}\right)\right) \] \[ y^{2}=\ln \
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• Use Laplace transform to solve
  6. \( y^{\prime \prime}+y=1 ; y(0)=6 . y^{\prime}(0)=0 \) 7. \( y^{\prime \prime}-4 y^{\prime}+4 y=\cos (t) ; y(0)=1 . y^{\prime}(0)=-1 \)
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• use Laplace transform to solve initial value
  8. \( y^{\prime \prime}+9 y=t^{2} ; y(0)=y^{\prime}(0)=0 \) 9. \( y^{\prime \prime}+16 y=1+r ; y(0)=-2, y^{\prime}(0)=1 \)
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• use Laplace transform to solve initial value
  \( y^{\prime}+4 y=1 ; y(0)=-3 \) \( y^{\prime}-9 y=t ; y(0)=5 \) \( y^{\prime}+4 y=\cos (t) ; y(0)=0 \)
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• AYUDA ME URGE :( Solucione con método de euler
  3. Utilizando el método de Euler calcula \( y(3) \). Considera \( \frac{d y}{d x}=3 x-2 y \) y que \( y(0)=\frac{3}{2} \), utiliza un tamaño de paso de \( h=1 \). 4. Utilizando el método de Euler c
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