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Unformatted text preview: MATH 310, Fall 2011: Differential Equations Simon Fraser University Quiz 5: Solutions 1. Consider the autonomous differential equation (of the form dy/dt = f ( y )) dy dt = 2 y 2 ( y 2- 9) . Sketch a graph of f ( y ) versus y , draw the phase line, find all the equilibrium points, and classify their stability. If y (0) = 1, what is lim t →∞ y ( t )? [Note: the phase line is simply the horizontal ( y ) axis of the graph of y = f ( y ) versus y , with clearly marked equilibria and arrows indicating where the solution is increasing and where it is decreasing.] Solution: A sufficiently complete answer is as follows: We have y = f ( y ) = 2 y 2 ( y + 3)( y- 3) . The fixed points (where y = f ( y ) = 0 ) are at y =- 3 , y = 0 (double zero of f ) and y = 3 . The graph of f ( y ) = y versus y is shown below; the arrows on the horizontal axis (the phase line) indicate where y ( t ) is increasing and where it is decreasing with t . We can read off stability from the graph: • y =- 3 is asymptotically stable • y = 0 is semi-stable (half-stable; it would also be correct to say it is unstable) • y = 3 is unstable Since y < for < y < 3 , and there is a fixed point at y = 0 , we find that if y (0) = 1 , then y ( t ) → as t → ∞ , that is, y (0) = 1 = ⇒ lim t →∞ y ( t ) = 0 . [A detained explanation of this answer is given after the solution to question 2.] 2. Find the value of b (if any) for which the following differential equation is exact, and find the general solution for that b : ( xy 2 + bx 2 y + e 2 x ) + ( ( x + y ) x 2 + 2 ) dy dx = 0 ....
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