hw2 - positive 6 Find the solution of the following initial...

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1. Find the solution of the initial value problem ¨ x - x = 0 , x (0) = 5 4 , ˙ x (0) = - 3 4 . Then determine the minimum value of the solution. 2. Solve the initial value problem ¨ x - ˙ x - 2 x = 0, x (0) = α , ˙ x (0) = 2. Then find α so that the solution approaches zero as t → ∞ . 3. Consider the equation ¨ x - (2 α - 1) ˙ x + α ( α - 1) x = 0 . Determine the values of α , if any, for which all solutions tend to zero as t → ∞ ; also determine the values of α , if any, for which all (nonzero) solutions become unbounded as t → ∞ . 4. Consider the following initial value problem: ¨ x - ˙ x + 1 4 x = 0 , x (0) = 2 , ˙ x (0) = b. Find the solution as a function of b and then determine the critical value of b that separates solutions that grow positively from those that eventually grow negatively. 5. Consider the initial value problem x + 12 ˙ x + 4 x = 0 , x (0) = a > 0 , ˙ x (0) = - 1 . (a) Solve the inital value problem. (b) Find the critical value of a that separates solutions that become negative from those that are always
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Unformatted text preview: positive. 6. Find the solution of the following initial value problem, sketch the graph of the solution, and describe its behavior for increasing t: ¨ x-2 ˙ x + 5 x = 0 , x ( π/ 2) = 0 , ˙ x ( π/ 2) = 2 . 7. Find the general solution of the following di²erential equation: 2¨ x + 3 ˙ x + x = t 2 + 3 sin t. 8. Find the solution of the following initial value problem: ¨ x + 2 ˙ x + 5 x = 4 e-t cos 2 t, x (0) = 1 , ˙ x (0) = 0 . 9. Solve the di²erential equation ¨ x + 2 ˙ x + 5 x = b 1 , ≤ t ≤ π/ 2, , t > π/ 2 with the initial conditions x (0) = 0 and ˙ x (0) = 0. Hint: ±rst solve the initial value problem for t ≤ π/ 2; then solve for t > π/ 2, determining the constants in the latter solution by assuming that x and ˙ x are continuous at t = π/ 2. 1...
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