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Calculus Archive: Questions from September 15, 2022

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  8. Next to each DE below, place the letters of all applicable properties. (a) Linear (b) Nonlinear (c) Separable (d) Homogeneous i) \( y^{\prime}=\frac{x^{2}-5 x y}{3 x y} \) ii) \( y^{\prime}=\frac{y
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  Find the General Solution \[ x y^{\prime \prime}+(2-x) y^{\prime}-y=0 \quad, \quad y(x)=\frac{1}{x} \]
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• i need help
  allar las derivadas de las siguientes funciones.(20 puntos) 1. Hallar la derivada de la función \( f(\mathrm{x})=\mathrm{x}^{2}-6 \mathrm{x}-5 \) en \( x=2 \). 2. Calcular la derivada de la función
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  2. If \( y=\left(2 x^{5}+7\right)^{\tan (x)} \), find \( d y / d x \).
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  2. Solve the IVP. \[ 5 y^{\prime \prime}+y^{\prime}=-6 x, \quad y(0)=0, \quad y^{\prime}(0)=-10 \]
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  3. Solve the IVP. \[ y^{\prime \prime}+4 y^{\prime}+4 y=(3+x) e^{-2 x}, \quad y(0)=2, \quad y^{\prime}(0)=5 \]
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  7. Solve the IVP: \( y^{\prime \prime \prime}+7 y^{\prime \prime}+16 y^{\prime}+12 y=0 ; y(0)=1, y^{\prime}(0)=0, y^{\prime \prime}(0)=0 \)
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  1) \( \left[7\right. \) pts.] Considera la ecuación diferencial \( y^{\prime}=y\left(y^{2}-4\right) \). a) Halla los puntos de equilibrio asociados a la ecuación diferencial. b) Clasifica la estabil
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• Find the solution of the differential equation
  La solución de la ecuación diferencial \( (x-y \ln y+y \ln x) d x+x(\ln y-\ln x) d y=0 \), es: a. \( (y-x) \ln x+y \ln y=c x+y \) b. \( (x-y) \ln x+y \ln y=c x+y \) c. \( (x-y) \ln x-y \ln y=c x+y \
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  2) Halla la solución de la E.D.O. \( \frac{d y}{d x}=\frac{1}{x+x y} \).
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  9) Given \( y=4 \cos (\theta)+10 \sin (\theta) \), find \( \left.\frac{d^{2} y}{d \theta^{2}}\right|_{\theta=\pi}=? \)
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• HELP PLEASE
  (2.) \( \left(4 y^{2}-3 x^{2}\right) d x+\left(8 x y+12 y^{2}\right) d y=0, y(0)=1 \) \( y^{\prime}+\left(\frac{2}{t}\right) y=\frac{\cos t}{t^{2}}, y(i 1)=0, t>0 \)
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  Find \( F^{\prime}(x) \) \[ F(x)=\int_{0}^{x^{5}} \sin t^{4} d t \]
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• Differential Equation
  \( (2 x+1) d x-(x+3 y+2) d y=0 \) \( (x+y-1)^{2}(4 x-6 y-1)^{3}=C \) \( (x+y+1)^{2}(4 x-6 y-1)^{3}=C \) \( (x+y+1)^{2}(4 x-3 y+1)^{3}=C \) \( (x+y+1)^{2}(4 x-6 y+1)^{3}=C \) \( (x+y-1)^{2}(4 x-6 y+1)^
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  (1 point) Find an explicit general solution for 1) \( y^{\prime}=\frac{9}{x} \Rightarrow y= \) 2) \( y^{\prime}=-6 \sin x+3 \cos x \Rightarrow y= \) \( +C \) 3) \( y^{\prime}=-9 e^{x} \Rightarrow y= \
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  (1 point) Perform the following integrations: 1) \( y^{\prime}=6 e^{6 x} \sin \left(e^{6 x}\right) \Rightarrow y= \) \( +C \) 2) \( y^{\prime}=\frac{x+1}{x^{2}+2 x+12} \Rightarrow y= \) 3) \( y^{\prim
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  Solve the initial value problem \[ y^{\prime \prime}+6 y^{\prime}+9 y=0, y(0)=2, y^{\prime}(0)=-7 \] \[ y=a e^{b t}+c t e^{d t} \] \[ a= \] \[ b= \]
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  1. (a) If \( 2 x \leq g(x) \leq x^{4}-x^{2}+2 \) for all \( x \), evaluate \( \lim _{x \rightarrow 1} g(x) \). (b) If \( \cos (x) \leq \frac{x}{\sin (x)} \leq \sec (x) \) for all \( x \) near 0 , eval
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  \( \int(12 \sin x+6 \tan x) d x \)
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  Find each indefinite integral Problem 1. \( \int x e^{3 x} d x \) Problem 2. \( \int \cos ^{2} \theta d \theta \)
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  d) \( y=e^{-x^{2}}, y=0 \) and \( x=1 \), about the \( y \)-axis.
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• with the steps, please
  Problem \( 2.6 .3 \). Find \( y^{\prime} \) If (i) \( y=-3 x^{2 \pi}+\frac{e^{\pi}}{3 \pi-1} \) (ii) \( y=3 x^{8}+2 x+1 \) (iii) \( y=\sqrt{x}+\frac{1}{x} \) \( (\mathrm{iv})=x^{e}+\frac{1}{x^{\sqrt{1
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