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Advanced Math Archive: Questions from December 19, 2022

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  Find the integrals over the unit circle \( \gamma \) : (a) \( \int_{\gamma} \frac{\cos z}{z} d z \) (b) \( \int_{\gamma} \frac{\sin z}{z} d z \) (c) \( \int_{\gamma} \frac{\cos \left(z^{2}\right)}{z}
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  7. A general solution to \( x^{2} y^{\prime \prime}+x y^{\prime}-9 y=0, x>0 \) is i) \( y=c_{1} x^{3} \) b) \( y=c_{1} x^{3}+c_{2} x^{-3} \) c) \( y=c_{1} \cos (\ln 3 x)+c_{2} \sin (\ln 3 x) \) d) \(
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  1. Find a general solution to the D.E. \( y^{(4)}+y^{\prime \prime \prime}+4 y^{\prime \prime}+4 y^{\prime}=4 x \)
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  10. The Wronskian of the D.E \( y^{\prime \prime}+\sin (x) y^{\prime}+y=0, x>0 \) is a) \( y=c e^{-\cos x} \) b) \( y=c e^{2 \sin x} \) c) \( y=c e^{\cos x} \) d) \( y=c e^{2 \cos x} \) e) \( y=c e^{-
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  \( \int_{-4}^{4} \int_{-10}^{10}\left(4-\left(\frac{x^{3}}{4000}-\frac{y}{40}\right)\right) d y d x \)
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  Solve the problem \[ \begin{array}{l} \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0,0
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  \( \int_{0}^{20} \int_{x}^{\sqrt{40 x-x^{2}}} 9 \sqrt{x^{2}+y^{2}} d y d x= \) \( \int_{0}^{20} \int_{x}^{\sqrt{40 x-x^{2}}} 9 \sqrt{x^{2}+y^{2}} d y d x= \)
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  Find a (real) general solut 1. \( y^{n}+4 y=\sin 3 x \)
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  11. If \( \tan \beta=\frac{\tan \alpha+\tan \gamma}{1+\tan \alpha \tan \gamma} \) and \( \sin 2 \beta=\frac{\sin 2 \alpha+\sin 2 \gamma}{\lambda+\sin 2 \alpha \sin 2 \gamma} \) then \( \lambda= \)
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• 21,22,23,24,25
  IVPs, SOME WITH DISCONTINUOUS INPUT Using the Laplace transform and showing the details, solve 18. \( 9 y^{\prime \prime}-6 y^{\prime}+y=0, \quad y(0)=3, y^{\prime}(0)=1 \) 19. \( y^{\prime \prime}+6
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  Solve the problem \[ \begin{array}{l} \frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0,0
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  \( \int_{31}^{62} \int_{0}^{41} \int_{y}^{41} \frac{x \cos z}{z} d z d y d x \)
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• 6. Show that the provided set of functions is orthogonal on the indicated interval.
  6. Muestre que el conjunto de funciones provisto es ortogonal en el intervalo indicado. a) \( f_{1}(x)=x ; f_{2}(x)=\cos (2 x) \); en \( [-\pi / 2, \pi / 2] \) b) \( \left\{\sin \left(\frac{n \pi}{p}
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  \( y^{\prime \prime \prime}+y^{\prime \prime}+3 y^{\prime}-5 y=16 e^{-x} \quad y(0)=0 \quad y^{\prime}(0)=2 \quad y^{\prime \prime}(0)=-4 \)
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  Find the critical values of the following functions and classify them. (a) \( y=x^{3} \) (b) \( y=x^{6}+5 \) (c) \( y=(x-1)^{3}+16 \) (d) \( y=(5-2 x)^{4}+8 \)
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  solve the 2 nd order differential equations: \[ \begin{array}{c} y^{\prime \prime}+4 y^{\prime}+3 y=0, \quad y(0)=2, y^{\prime}(0)=-1 \\ 5 y^{\prime \prime}+2 y^{\prime}+7 y=0, \quad y(0)=2, y^{\prime
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  Evaluate \( \int_{39}^{80} \int_{0}^{4} \int_{y}^{4} \frac{x \cos z}{z} d z d y d x \) \( x \)
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  Solve the initial value problem \[ y^{\prime \prime}+9 y=-12 \sin (3 x), \quad y(0)=\frac{8}{5}, y^{\prime}(0)=8 \] \( ? \)
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  Solve the initial value problem \[ y^{\prime \prime}+9 y=-12 \sin (3 x), \quad y(0)=\frac{8}{5}, \quad y^{\prime}(0)=8 \] \[ y(x)= \]
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  4) Solve the following first-order partial DEs: (a) \( (y-u) u_{x}+(x-y) u_{y}=u-x \) Ans: \( \frac{1}{2} x^{2}+y u=\phi(x+y+u) \). (b) \( x^{2} u_{x}+y^{2} u_{y}=(x+y) u \) Ans: \( \phi\left(\frac{x
  2 answers