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Advanced Math Archive: Questions from March 29, 2023

• Use the Laplace transform to solve the initial value problem.
  18. \( y^{\prime \prime}-4 y^{\prime}+4 y=4 e^{2 t}, y(0)=-1, y^{\prime}(0)=-4 \). 19. \( y^{\prime \prime}-y=4 \cos t, y(0)=0, y^{\prime}(0)=1 \). 20. \( y^{\prime \prime}-2 y^{\prime}-4 y=-2, y(0)=1
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  17. \( y^{\prime \prime}-2 y^{\prime}+y=t e^{t}+4, \quad y(0)=1, \quad y^{\prime}(0)=1 \)
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• solve using Laplace transform
  \[ y^{\prime}(0)=-5 \] 21. \( y^{\prime \prime}+4 y=4 \cos t \), if \( 0
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• Calcular las 4 derivadas parciales de segundo orden
  3. Calcular las cuatro derivadas parciales de segundo orden \( z=f(x, y)=x^{4}-3 x^{2} y^{2}+y^{4} . \quad \) Se cumple: \( \frac{\partial^{2} f}{\partial x \partial y}=\frac{\partial^{2} f}{\partial
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  30) \( \begin{aligned} y^{\prime \prime}(\theta)+2 y^{\prime}(\theta)+y(\theta) & =2 \cos \theta: \\ y(0)=3, \quad y^{\prime}(0) & =0\end{aligned} \)
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  1. Demuestra que si \( f \) es continua y \( B \) es boreliano entonces \( f^{-1}(B) \) es boreliano.
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  1. Solve the following ODE's, also determine the basis of \( y_{h}(x) \) (a) \( y^{\prime \prime}+5 y^{\prime}+6 y=10 e^{2 x} \) (b) \( y^{\prime \prime}+4 y=4 e^{-2 x} \) (c) \( y^{\prime \prime}+4 y
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  (1 point) Find \( y \) as a function of \( x \) if \[ y^{\prime \prime \prime}-16 y^{\prime \prime}+63 y^{\prime}=144 e^{x} \] \[ \begin{array}{l} y(0)=28, \quad y^{\prime}(0)=17, \quad y^{\prime \pri
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• Needs answer please ??
  \( y^{\prime}=-12 x y ; y=4 e^{-6 x^{2}} \)
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• question 5
  Solve the IVPs below using the Laplace transform: 1. \( y^{\prime}-3 y=2^{2 t} ; y(0)=1 \). 2. \( y^{\prime \prime}-6 y^{\prime}+9 y=t^{2} e^{3 t} ; y(0)=2, y^{\prime}(0)=6 \). 3. \( y^{\prime \prime}
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  Use \( f(x, y, z)=x^{2}+y z, \vec{F}(x, y, z)=\langle x y, y z, x z\rangle \), and \( \vec{G}(x, y, z)=\left\langle-\sin (z), e^{x z}, y\right\rangle \). Compute \( (\vec{F} \times \vec{G})(7,-1,5) \)
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  homogéneas, exactas o ninguna de estas: 1. Exacta: 2. Homogénea: 3. Ninguna de las anteriores: a. \( (x+1) \frac{d y}{d x}=\frac{x-y}{x+1} \) b. \( (x+1) \frac{d y}{d x}=10-y \) c. \( \frac{d y}{d
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  II. Resuclva la ecuación Diferencial a. Resuelva la ecuación diferencial \( (x-y) d x+x d y=0 \) b. Resuelva la ecuación diferencial \( \left(x^{2}+y^{2}\right) d x+\left(x^{2}-x y\right) d y=0 \)
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• Solve
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  \( \begin{array}{r}c=x-4 y \text { subj } \\ 3 x+y \geq 5 \\ 2 x-y \geq 0 \\ x-3 y \leq 0 \\ x \geq 0, y \geq 0\end{array} \)
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  \( y^{\prime \prime}+4 y=f(x) ; \quad y(0)=1, y^{\prime}(0)=0 \quad \) where \[ f(x)=\left\{\begin{array}{ll} 0 & 0 \leq x
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  Give the general solution of the differential equation a) \( y=c_{1} \sin (3 x)+c_{2} \cos (3 x)+\frac{1}{3} x \sin (3 x)-\frac{1}{3} \cos (3 x) \) b) \( y=c_{1} e^{3 x}+c_{2} e^{-3 x}-\frac{1}{3} x \
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• Determine the limit, if it does not exist, explain why. a) lim(x,y)-(0.1) x=y b) lim(x,y)-(0,0) √x+√5 arccos(*/y) 1+xy 3 c) lim(x,y) (0,0) x²y² II. Analyze the continuity of the function a) f(x,
  1. Determine el limite, en caso de que no exista explique por qué. a) \( \lim _{(x, y) \rightarrow(0,1)} \frac{\arccos (x / y)}{1+x y} \) b) \( \lim _{(x, y) \rightarrow(0,0)} \frac{x-y}{\sqrt{x}+\sq
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  Solve the following ODE's, also determine the basis of \( y_{h}(x) \) (a) \( y^{\prime \prime}+5 y^{\prime}+6 y=10 e^{2 x} \) (b) \( y^{\prime \prime}+4 y=4 e^{-2 x} \) (c) \( y^{\prime \prime}+4 y^{\
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  3. Find the derivative of \( y \) in each case. (a) \( y=\frac{\sin (x)}{\cos (x)} \) (b) \( y=\sin ^{3}(x) \) (c) \( y=x \cdot \sin \left(x^{3}\right) \)
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• De acuerdo a su naturaleza, la funcion f(x)=e^(x+6) es del tipo:
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