1005 test 1 old test

1005 test 1 old test - 2 xy Answer y = kx 2 e x/y[6 5 Find...

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MATH 1005B Test 1 Winter 2009 1 This test paper has two parts and total of 30 marks. Part I has 3 multiple choice questions. Part II has 4 long answer questions. It cannot be taken from the examination room. Only nonprogrammable calculators are allowed. Duration: 50 Minutes. NAME : STUDENT NO : PART I: Multiple Choice Questions. No partial marks. Circle the correct answer. [2] 1. If y is the solution of the initial value problem y ± + 1 x y =4 x 2 ,y (1) = 3, what is y (2) ? (a) - 8 (b) 8 (c) 9 (d) 3 (e) 2 [2] 2. If y is the solution of the initial value problem y ± = e x 2 y y (0) = 2, what is y (1) ? (a) e (b) e + 1 (c) e +3 (d) e + 3 (e) e +1 [2] 3. What is the family of orthogonal trajectories for the curves y = ke 2 x ? (a) y 2 + x 2 = c (b) y 2 + x = c (c) y + x 2 = c (d) y + x = c (e) y + e - 2 x = c PART II: Long answer questions. Show all your work. [6] 4. Find the general solution of the differential equation y ± = xy +2 y 2 x
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Unformatted text preview: 2 + xy . Answer: y = kx 2 e x/y [6] 5. Find the general solution of the differential equation y ±-y = xy 2 . Answer: y = 1-x + 1-c e-x [6] 6. Find the general solution of the differential equation (4 x 3 +3 x 2 y +3 y 2 )+( x 3 +6 xy ) y ± = 0. Answer: x 4 + x 3 y + 3 y 2 x = K [6] 7. Show that the differential equation (3 xy +2 y 2 )+( x 2 +2 xy ) y ± = 0 is not exact, and ±nd an integrating factor which makes it exact. Write down the new exact differential equation, but do not solve it. Answer: P y ( x, y ) = 3 x + 4 y, Q x ( x, y ) = 2 x + 2 y. Hence P y ( x, y ) ± = Q x ( x, y ) . I ( x ) = x (Integrating factor) . (3 x 2 y + 2 xy 2 ) + ( x 3 + 2 x 2 y ) y ± = 0 (New exact differential equation) ....
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