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
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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## This note was uploaded on 01/12/2010 for the course MATH 42 at Stanford.

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

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