Question #283799

a) A thin metal plate bounded by the π‘₯-axis and the lines π‘₯ = 0 and π‘₯ = 1 and


stretching to infinity in the 𝑦-direction has its upper and lower faces perfectly


insulated and its vertical edges and edge at infinity are maintained at the constant


temperature 0 °𝐢, while over the base temperature of 50 °𝐢 is maintained. Find


the steady state temperature 𝑒(π‘₯,𝑑).


[5]


b) If 𝑓(π‘₯𝑦^2, 𝑧 βˆ’ 2π‘₯) = 0 then prove that π‘₯ 𝑧x βˆ’1/2𝑦 𝑧𝑦 = 2 π‘₯

1
Expert's answer
2022-01-05T14:02:51-0500

b)

prove:

xβˆ‚zβˆ‚xβˆ’βˆ‚zβˆ‚yy2=2xx\frac{\partial z}{\partial x}-\frac{\partial z}{\partial y}\frac{y}{2}=2x


Differentiating 𝑓(π‘₯𝑦2,π‘§βˆ’2π‘₯)=0𝑓(π‘₯𝑦^2, 𝑧 βˆ’ 2π‘₯) = 0 , we get:

y2f1+(βˆ‚zβˆ‚xβˆ’2)f2=0y^2f_1+(\frac{\partial z}{\partial x}-2)f_2=0


2xyf1+βˆ‚zβˆ‚yf2=02xyf_1+\frac{\partial z}{\partial y}f_2=0


where f1 and f2 are partial derivatives of f with respect to its first input and second input. When we multiply the first equality above by 2x and the second by y:

2x(βˆ‚zβˆ‚xβˆ’2)=yβˆ‚zβˆ‚yβ€…β€ŠβŸΉβ€…β€Šxβˆ‚zβˆ‚xβˆ’βˆ‚zβˆ‚yy2=2x2x(\frac{\partial z}{\partial x}-2)=y\frac{\partial z}{\partial y}\implies x\frac{\partial z}{\partial x}-\frac{\partial z}{\partial y}\frac{y}{2}=2x


a)

solution of two dimensional heat equation βˆ‚2uβˆ‚x2+βˆ‚2uβˆ‚y2=0\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0 is


u(x,y)=(AcosΞ»x+BsinΞ»x)(CeΞ»y+Deβˆ’Ξ»y)u(x,y)=(Acos\lambda x+Bsin \lambda x)(Ce^{\lambda y}+De^{-\lambda y})

boundary conditions:

u(x,∞)=0u(x,\infin)=0

u(x,0)=50u(x,0)=50

u(0,y)=0u(0,y)=0

u(1,y)=0u(1,y)=0

0≀x≀1,0≀y<∞0\le x\le 1,0\le y< \infin


then:

u(0,y)=A(CeΞ»y+Deβˆ’Ξ»y)=0β€…β€ŠβŸΉβ€…β€ŠA=0u(0,y)=A(Ce^{\lambda y}+De^{-\lambda y})=0\implies A=0

u(x,y)=BsinΞ»x(CeΞ»y+Deβˆ’Ξ»y)u(x,y)=Bsin \lambda x(Ce^{\lambda y}+De^{-\lambda y})


u(1,y)=BsinΞ»(CeΞ»y+Deβˆ’Ξ»y)=0β€…β€ŠβŸΉβ€…β€ŠΞ»=nΟ€u(1,y)=Bsin \lambda (Ce^{\lambda y}+De^{-\lambda y})=0\implies \lambda=n\pi

then:

u(x,y)=Bsin(Ο€nx)(CeΟ€ny+Deβˆ’Ο€ny)u(x,y)=Bsin (\pi nx)(Ce^{\pi n y}+De^{-\pi n y})

u(x,∞)=Bsin(Ο€nx)CeΟ€ny=0β€…β€ŠβŸΉβ€…β€ŠC=0u(x,\infin)=Bsin (\pi nx)Ce^{\pi n y}=0\implies C=0


u(x,y)=B1sin(Ο€nx)eβˆ’Ο€nyu(x,y)=B_1sin (\pi nx)e^{-\pi n y}

where B1 = BD


the most general solution:

u(x,y)=βˆ‘Bnsin(Ο€nx)eβˆ’Ο€nyu(x,y)=\sum B_nsin (\pi nx)e^{-\pi n y}

where

Bn=∫01u(x,0)sin(Ο€nx)=50∫01sin(Ο€nx)=B_n=\int^1_0 u(x,0)sin (\pi nx)=50\int^1_0 sin (\pi nx)=


=βˆ’50Ο€ncos(nΟ€x)∣01=50Ο€n(1βˆ’cos(nΟ€))=-\frac{50}{\pi n}cos(n\pi x)|^1_0=\frac{50}{\pi n}(1-cos(n\pi))


for odd n:

u(x,y)=βˆ‘100Ο€nsin(Ο€nx)eβˆ’Ο€nyu(x,y)=\sum\frac{100}{\pi n}sin (\pi nx)e^{-\pi n y}


for even n:

u(x,y)=0u(x,y)=0


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