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UMD PDE Qualifying Exams/EvansCh2

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Problem 1

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Write an explicit formula for a function solving the initial-value problem


where and are constants.

Solution

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Consider characteristics . Also, for any , consider . Then taking a derivative gives

where the last inequality is a result of the original PDE. The above ODE can be solved and we get

Finally, any point is connected to the characteristic curve where and hence

.