This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 201 Fall 2011 Midterm Exam 1 (A) October 4, 2011 1. [4 marks] Does the differential equation x dy dx = y 2 , y ( x ) = 1 necessarily have a unique solution in some small interval ( x h,x + h ) for the given values of x ? Why? (a) x = 1 ; (b) x = 0 . • The existence and uniqueness theorem has two conditions to check: f ( x,y ) = y 2 /x is continuous, and ∂f/∂y = 2 y/x is continuus • for (a), near (1 , 1) , both f and ∂f/∂y are continuous. So there is a unique solution • for (b), neither f nor ∂f/∂y is continuous at (0 , 1) , the theorem cannot be applied. So it does not necessarily have a unique solution 2. [5 marks] Sketch the phase portrait of the autonomous equation dN dt = N 2 1 and classify each equilibrium point as stable, unstable, or semistable. • Equilibria: N 2 1 = 0 , N = ± 1 • These equilibria cut the phase space (the real line) into three intervals (∞ , 1) , ( 1 , 1) , (1 , ∞ ) • In each interval – (∞ , 1) and (1 , ∞ ) , N > , solution curves increase (moving to the right)...
View
Full
Document
This note was uploaded on 01/15/2012 for the course MATH 201 taught by Professor Steacy during the Winter '10 term at University of Victoria.
 Winter '10
 STEACY
 Math

Click to edit the document details