m257_316_midterm1_sec101w2006

m257_316_midterm1_sec101w2006 - x ) y = 0 (1) (a) Classify...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 257/316, Midterm 1, Section 101 11 October 2006 Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed. Maximum score 100. 1. Apply the method of separation of variables to determine a solution to the one dimensional heat equation with homogeneous Neumann boundary conditions, i.e. u t = α 2 2 u x 2 BC : u (0 ,t ) x =0= u ( L, t ) x IC : u ( x, 0) = f ( x ) Evaluate the coe cients of the corresponding Fourier series for f ( x )= x and L =1 . Hence determine the solution of the above initial-boundary value problem. [40 marks] 2. Consider the second order di f erential equation: Ly =2 x 2 y 00 3 xy 0 +(3+
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: x ) y = 0 (1) (a) Classify the points x (do not include the point at in f nity) as ordinary points, regular singular points, or irregular singular points. [5 marks] (b) Explain how you would classify the point at in f nity (you need not carry out the calculations). [5 marks] (c) Explain how you would obtain a general solution of (1) about the point x = 1 . [10 marks] (d) Use the appropriate series expansion about the point x = 0 to determine two independent solutions to (1). Determine the radius of convergence of one of these series. [40 marks] 1...
View Full Document

Ask a homework question - tutors are online