20d_07s_1e

# 20d_07s_1e - y ′′ p t y ′ q t y = 0 How can you tell...

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Math 20D Midterm Exam (50 points) Friday 5/4/2007 Please put your name and ID number on your blue book. CLOSED BOOK, but ONE SIDE of one page of notes are allowed. Calculators are NOT allowed. In a multipart problem, you can do later parts without doing earlier ones. You must show your work to receive credit. 1. (10 pts.) Here are some di±erential equations for the function y . For each equation (i) give its order and (ii) tell whether or not it is linear. (a) y ′′ ( t ) = t 2 y ( t ) + 7 (b) ( x 2 + 1) dx = ( x + 1) dy (c) ( y 2 ) + y = 1 (d) y y ′′ = 2 (e) x 2 y ( x ) + xy ( x ) + x 3 = 0 2. (2 pts.) The functions p ( t ) and q ( t ) are continuous for all t and y 1 and y 2 are particular solutions to the linear homogeneous equation
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Unformatted text preview: y ′′ + p ( t ) y ′ + q ( t ) y = 0. How can you tell if c 1 y 1 + c 2 y 2 is the general solution? 3. (6 pts.) Find the critical points (also called equilibrium points) of the autonomous di±erential equation dy/dt = y (1-y 2 ) and classify each one as asymptotically stable or unstable. 4. (32 pts.) Solve each of the following di±erential equations. If no initial conditions are given, ²nd the general solution. (a) y ′′ + 9 y = 0; y (0) = 0, y ′ (0) = 6. (b) dx/dt = e x + t ; x (0) = 1. (c) (2 x + y ) dx + ( x-2 y ) dy = 0. (d) ty ′ ( t )-y ( t ) = t 2 , t > 0. END OF EXAM...
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