Mth3006 Weekly Problems 6

Question One

Use separation of variables to solve the following differential equation:

$$\frac{\partial u}{\partial x}=4\frac{\partial u}{\partial y}:u(0,y)=8e^{-3y}+4e^{-5y}$$

Solution

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Question Two

Solve the following boundary value problems by separation of variables…

Part A

$$3\frac{\partial u}{\partial x}+2\frac{\partial u}{\partial y}=0:u(x,0)=4e^{-x}$$

Solution

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Part B

$$\frac{\partial u}{\partial x}=2\frac{\partial u}{\partial y}+u:u(x,0)=3e^{-5x}+2e^{-3x}$$

Solution

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Question Three

Solve the following heat equation problem by separation of variables:

$$\frac{\partial u}{\partial t}=\frac{{\partial }^{2}u}{\partial x^2}$$

with

$$\frac{\partial u(0,t)}{\partial x}=0,u(2,t)=0,u(x,0)=8\cos \left(\frac{3\pi x}{4}\right)-6\cos \left(\frac{9\pi x}{4}\right)$$

Solution

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Question Four

Solve the following wave equation problem by separation of variables:

$$\frac{{\partial }^{2}y}{\partial t^2}=4\frac{{\partial }^{2}y}{\partial x^2}$$

with

$$y(0,t)=y(5,t)=0,y(x,0)=0,\frac{\partial y(x,0)}{\partial t}=5\sin (\pi x)$$

Solution

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