MATH 285 E1 F1 GRADED HOMEWORK SET 6 DUE FRIDAY NOVEMBER 21 IN LECTURE This time the homework has just one part Please staple your homework together and put your name and section on it Thank you 1 15 points Consider the eigenvalue problem 00 0 y 2y y 0 y 0 0 y 1 0 Find the eigenvalues and eigenfunctions for this problem That is find the values of for which the problem has a nontrivial solution and find those nontrivial solutions Hint The smallest eigenvalue is 2 1 with associated eigenfunction e x sin x In your answer you should verify this 2 10 points Find the solution of the heat problem on the interval 0 x 5 u 2 2 xu2 t u 0 t 0 x u 5 t 0 x u x 0 cos2 10 x 3 15 points Find the solution of the heat problem on the interval 0 x 1 u 2 5 xu2 t u 0 t 0 u 1 t 1 u x 0 x2 You should use the following strategy a Find a steady state solution u0 of all parts of the problem except the initial condition That is find a function u0 x t that is constant in time u t 0 that satisfies the heat equation and that satisfies the conditions u0 0 t 0 and u0 1 t 1 b Posit w u u0 and show that w must solve a slighty different heat problem w 2 5 xw2 t w 0 t 0 w 1 t 0 w x 0 x2 u0 x 0 1 2 GRADED HOMEWORK 6 w is called the transient term c Using the methods described in the lectures find the solution w x t and hence u x t
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