12:49 Wed 7 Aug
\( 39 \% \)
Done
MTH6151 Late Summer Exam (6 of 8)
MTH6151 / MTH6151P (Late Summer 2024)
Page 5
Question 4 [14 marks]. Suppose \( U_{1}(r, \theta) \) and \( U_{2}(r, \theta) \) are two solutions to the Laplace equation in the disk of radius 1 .
(a) Is \( U_{3} \) defined by \( U_{3}(r, \theta)=U_{1}(r, \theta)-2 U_{2}(r,-\theta) \) also a solution to the Laplace equation on the disk of radius 1 ? Explain your answer.
\( [6] \)
(b) If \( U_{1}(1, \theta) \geq U_{2}(1, \theta) \) for any \( \theta \), show that \( U_{1} \geq U_{2} \) in the whole disk of radius 1 without solving the equation.
(c) Suppose \( U_{1} \) has the following boundary condition
\[
U_{1}(1, \theta)=-4 \cos \theta+2 .
\]
What is the value of \( U_{1} \) at the center of the disk?
[4]
