GSW Problem Set 1 - y is now odd rather than even Determine...

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EA4 Workshop questions set 1 - week of 9/29/08 1. Consider the equation dy dt = y n , y (0) = 1 , (1) where n is a positive integer. (a) Find the solution for all values of n 1 . (b) Show that for n 2 the solution blows up at a finite time. (c) Show that the blowup time gets smaller as n increases. Explain why this is so. (d) Explain why blowups are reasonable from the equation. 2. Consider the equation dy dt = y 2 , y (0) = - 1 . Note, that this is different from the previous case in that the initial value of y is negative. (a) Determine the qualitative behavior of the solution graphically. Then find the solution and discuss the difference between this case and the case when y (0) = 1 . Explain this behavior. (b) Consider the equation dy dt = y 3 , y (0) = - 1 . Note, that this is different from the previous case in that the exponent of
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Unformatted text preview: y is now odd rather than even. Determine the qualitative behavior of the solution graphically. Then ±nd the solution and explain its behavior. (c) Explain why the behavior now depends on whether n is even or odd without referring to the exact solution. 3. Consider the two equations y = y + 1 , y (0) = 1 , (2) and z = z + 1 , z (5) = 1 . (3) Note that the solutions y ( t ) and z ( t ) satisfy the same equation. Only the initial condition differs. Without ±nding the solution show that z ( t ) = y ( t-5) . 4. Suppose we now consider the two equations y = y + t, y (0) = 1 , (4) and z = z + t, z (5) = 1 . (5) Can you still say that z ( t ) = y ( t-5) ?...
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