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**Unformatted text preview: **Math 251 May 2, 2005 Final Exam ANSWER KEY 1. Consider the following nonlinear system of ODEs: x = x- xy y = xy- 2 y a. 2pt Determine its critical points. ANS. Factoring gives = x ( y- 1) = y (2- x ) gives two critical points x = ! and x = 2 1 ! . b. 10pt Approximate this nonlinear system near each one of its critical points by a linear system of ODEs. State the type and stability of the critical point of approximating linear system. ANS. At ! the linearization is x =- x y = 2 y which has eigenvalues r 1 =- 1 and r 2 = 2, opposite signs. The corresponding eigenvectors are 1 ! and 1 ! . Hence the origin is a saddle for for this system, which is unstable, and the system has trajectories along the x-axis moving towards the origin and trajectories along the y-axis moving away the origin. At x = 2 1 ! we set u = x- 2 v = y- 1 . Substituting this into the original equation gives u = ( u + 2) v 2 v v = ( v + 1)(- u ) - u The eigenvalues of this system 2 are purely imaginary. Thus this critical point is a center for the approximating linear system. The elliptical trajectories are traced in the clockwise direction. c. 3pt Sketch a phase portrait near each of the critical points. (2 pt for finding each the critical pt). At ! : (2 pt for linearization at origin) (2 pt for the eigenvalue eigenvector pairs) (2 pt for the words Saddle and Unstable) At 2 1 ! (1 pt for definition of u and v ) (4 pt for linearization at the second critical point) (2 pt for the words Center and Stable) (1pt for sketching saddle at origin) (1pt for sketching ellipse at 2 1 ! ) (1pt for getting both orientations.) 2. Consider the function f ( x ) = if- 2 x < 3 if x < 1 if 1 x 2 a. 12pt Find the Fourier series of f ( x ) on [- 2 , 2]. (You may either write the first seven terms or use summation notation to write the entire series.) ANS. a = 1 2 Z 2- 2 f ( x ) dx = 1 2 Z 1 3 dx = 3 2 a n = 1 2 Z 2- 2 f ( x ) cos n 2 x dx = 3 2 Z 1 cos n 2 x dx = 3 2 2 n sin n 2 x 1 = 3 n sin 3 n 2 b n = 1 2 Z 2- 2 f ( x ) sin n 2 x dx = 3 2 Z 1 sin n 2 x dx =- 3 2 2 n cos n 2 x 1 = 1 n 1- cos n 2 3 4 + X n =1 3 n sin 3 n 2 cos n 2 x + 1 n 1- cos n 2 sin n 2 x (1 pt for finding a ) (4 pt for finding a n ) (4 pt for finding b n ) (1 pt for placing a / 2 in the final answer.) (2 pt for placing a n , b n in front of cos ( n 2 x ) . and sin ( n 2 x ) .) in final answer) b. 3pt Find lim n s n (9) where { s n ( x ) } denotes the sequence of partial sums for the Fourier series found in Part a . ANS. 3 2 . (3 pt, all or nothing) 3. a. 12pt Find the sine series of the function f ( x ) = x on the interval [0 , 3] (You may either write the first four terms or use summation notation to write the entire series.) ANS....

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