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Unformatted text preview: ODE, Review for Exam 1 June 9, 2008 1. Consider the autonomous equation x = x 3- x . (a) Sketch the phase line (the x axis) and the corresponding direction field on the plane. (b) Sketch graphs of some solutions. Be sure to include at least one solution with values in each interval above, below, and between critical points. (a) Suppose x ( t ) is a nonconstant solution to the above equation. If x 00 (2) = 0 what is x (2)? 2. Consider the equation x 00 + 2 x + 2 x = e- t cos 2 t . (a) Find a particular solution. (b) Find the general solution. 3. (a) If e- 2 t + 2 e- t solves x 00 + cx + kx = 0 what are c, k ? (b) Same if te- t instead. 4. Solve y 00- 4 y = e 2 t . 5. Consider the equation y = ty 2 . (a) Find the general solution. (dont forget lost solutions). (b) All but one solution blows up in finite time. Which solution doesnt? At what time does the solution satisfying x (0) = 1 blow up?...
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