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The problems and solutions for exam 1 of math 110, focusing on the transport equation, wave equation, and heat equation. Students are required to demonstrate their work for full credit. Problems include finding specific solutions to transport and wave equations, computing energy at a certain time for the wave equation solution, and showing that one solution is less than or equal to another for the heat equation.
Typology: Exams
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Please answer the following questions. Because this test is open book and open note, you will not get credit for answers unless you demonstrate how you arrived at them. In short, please show all work.
Problem 1. Please find the specific solution to the transport equation: 1 2 + sin(x)
∂xu + (y + 1)∂y u = 0 ,
with the initial data at y = 0:
u(x, 0) = cos(x) − 2 x.
1
2
Problem 2. For this problem, let u(x, t) be the specific solution to the wave equation: utt = uxx , on −∞ < x < ∞ u(x, 0) = 0 , ut(x, 0) = e−x
2 .
Please answer the following:
a) Compute the energy at time t = 100 for this solution E(100). Please explain your answer carefully.
b) Is it true that the solution u(x, t) is always strictly positive for 0 < t? That is, is it true that 0 < u(x, t) for all 0 < t.
