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M251Hfinal_mockup02(sp09)

M251Hfinal_mockup02(sp09) - u t = 9 u xx ≤ x ≤ 2 u x(0...

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MATH251H Practice Exam II Spring 2008 120 minutes 1. (15 points) Solve the following initial value problem, x = parenleftBigg 2 - 1 1 4 parenrightBigg x , x (0) = parenleftBigg 3 1 parenrightBigg . 1
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2. (15 points) Find the general solution of x = parenleftBigg 2 3 3 2 parenrightBigg x . Classify the type and stability of the critical point at (0,0).
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3. (20 points) Consider the nonlinear system, braceleftBigg x = x - y y = ( x - 1)( y - 2) . a. Find all the critical points. b. Linearize the system around each critical point, and classify its type and stability.
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4. (20 points) Consider the following nonlinear system: braceleftBigg x = - x 3 + xy 2 y = - 2 x 2 y - y 3 Is the critical point at the origin stable? (Hint: Consider the Liapunov function
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Unformatted text preview: u t = 9 u xx , ≤ x ≤ 2 , u x (0 , t ) = 0 u x (2 , t ) = 0 u ( x, 0) = cos( 3 π 2 x )-cos( πx ) . 6. (20 points) Let f ( x ) = b 2-x ≤ x ≤ 2 , 2 + x-2 ≤ x ≤ . a. Compute the Fourier coefcients o± f ( x ) ±or-2 ≤ x ≤ 2. b. Solve the initial-boundary value problem, u tt = 4 u xx , ≤ x ≤ 2 , u (0 , t ) = 0 u (2 , t ) = 0 u ( x, 0) = f ( x ) u t ( x, 0) = 0 . Express the solution as a Fourier series....
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