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UMD PDE Qualifying Exams/EvansCh2
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Problem 1
Write an explicit formula for a function solving the initial-value problem
where and are constants.
Solution
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
.