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DE4spring08

# DE4spring08 - MTH 2201 Differential Equations Homework 4...

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Unformatted text preview: MTH 2201 Differential Equations Homework 4 : Second Order Linear ODE Spring2008 1. Find all solutions of y + 2iy + y = 0 2. Find all solutions of y + (3i - 1)y - 3iy = 0 3. Find the general solution of (i) 4y + y = 0. (ii) y + 8y + 16y = 0. (iii) 8y + 2y - y = 0. (iv) y - y = 0 dy d5 y =0 (v) y - 4y - 5y = 0 (vi) 5 - 16 dx dx 4 2 dy dy (vii) 16 4 + 24 2 + 9y = 0. dx dx 4. Consider y + y - 6y = 0. (i) Compute the solution satisfying (0) = 1, (0) = 0. (ii) Compute the solution satisfying (0) = 0, (0) = 1. 5. Find all solutions of y + y = 0 satisfying (0) = 1, (/2) = 2 6. Let be a solution of the equation y + a1 y + a2 y = 0, where a1 , a2 are constants. If (t) = e(a1 /2)t (t). Show that satisfies the DE y + ky = 0, where k is some constant. 7. Find the solutions to the following IVPs : (i) y + (3i - 1)y - 3iy = 0, y(0) = 2.y (o) = 0. (ii) y + 10y = 0, y(0) = , y (0) = 2 . 8. Suppose is a function having a continuous derivative on 0 x < such that (x) + 2(x) 1 for all such x, and (0) = 0. Show that (x) < 1/2 x 0. 9. Determine the values of , for whcih all solutions of y - (2 - 1)y + ( - 1)y = 0, tend to zero as t 0. 1 10. An equation of the form t2 y + ty + y = 0, t > 0, is called an Euler Equation. Show that x = ln t, transforms the equation to an ODE with constant coefficients. Use this result to solve, for t > 0, t2 y + 3ty + 54y = 0. You may also solve the following problems from your text. Exercise 3.4 : Problems 1, 7, 13, 17, 21, 35, 37, 41, 43, 53, 55. Exercise 3.5 : 5, 15, 21, 23, 25, 29, 33, 35, 37, 39. 2 ...
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DE4spring08 - MTH 2201 Differential Equations Homework 4...

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