chap7pracB

chap7pracB - Math 42 Chapter 7 Practice Problems Set B 1....

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Math 42 — Chapter 7 Practice Problems — Set B 1. Which of the following functions is a solution of the differential equation dy dx = 4 xy ? (a) y = e - 4 x (b) y = 4 x (c) y = e 2 x 2 (d) y = - 4 x (e) y = e 2 x (f) y = 2 x 2 (g) y = 4 e 2 x 2 (h) y = 2 e 4 x 2. Which of the following functions are solutions of the differential equation y 00 + y 0 = 6 y ? Show your work. (a) y = e - 4 x (b) y = e - 3 x (c) y = e - 2 x (d) y = - 4 e - 2 x 2 (e) y = e 2 x (f) y = 2 x 2 (g) y = 2 sin(2 x ) (h) y = 4 e - 4 x (i) y = 3 e - 3 x (j) y = 3 e 2 x Direction Fields 3. Match each of the slope fields below with exactly one of the differential equations. (The scales on the x - and y -axes are the same.) Also, provide enough explanation to show why no other matches are possible. (i) y 0 = xy + 1 (ii) y 0 = sin x (iii) y 0 = xe - y (iv) y 0 = y 2 + 1 (v) y 0 = sin y (a) (b) (c) (d)
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4. The slope field for the differential equation y 0 = 0 . 5(1 + y )(2 - y ) is shown below. (The scales on the x - and y -axes are the same, and some of the values from - 2 to 2 are marked on the y -axis.) (a) For which regions are all solution curves increasing? Justify your answer using the differential equation. (b) For which regions do the solution curves tend toward a finite y -value as x → ∞ ? Justify your answer using the differential equation. 5. A direction field is given in the picture below. Which of the following represents its differential equation? Explain why each of the other differential equations is not represented by the direction field. (a) y 0 = y - x (b) y 0 = y 2 - x 2 (c) y 0 = y + x (d) y 0 = y 2 + x 2 (e) y 0 = y - x 2 (f) y 0 = x - y 2
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Separation of Variables 6. Solve the differential equation dy dx = x + sin x 3 y 2 . 7. Solve the initial value problem dy dx = 4 - 7 y, y (0) = 3 . Show all of your work, with full mathematical justification. 8. In this problem, we will solve the differential equation xy 0 + 2 y = cos( x 2 ), even though it is not a separable equation. (a) Suppose y ( x ) satisfies the above equation (for x 6 = 0). Verify that the new function z ( x ) = x 2 y ( x ) satisfies z 0 = x cos( x 2 ). (b) Use separation of variables to find all solutions to z 0 = x cos( x 2 ). (c) Solve the initial value problem xy 0 + 2 y = cos( x 2 ) , y ( π ) = 0 . (Hint: remember, the function x 2 y ( x ) is a solution to part (b).) 9. An equation used to model the growth of animal tumors is given by y 0 = - ay ln( y/b ), where a and b are positive constants. (This is known as the Gompertz equation.) (a) Find any equilibrium solutions of the Gompertz equation. (b) If
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chap7pracB - Math 42 Chapter 7 Practice Problems Set B 1....

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