# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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36. Assume y(t) = A(t)exp(-J (-2)dt) = A(t)e2t.

Differentiating y(t) and substituting into the D.E.

yields A' (t) = t2 since the terms involving A(t) add to zero. Thus A(t) = t3/3 + c, which substituted into y(t) yields the solution.

dt

37. y(t) = A(t)exp(-J ) = A(t)/t.

t

Section 2.2, Page 45

Problems 1 through 20 follow the pattern of the examples worked in this section. The first eight problems, however, do not have I.C. so the integration constant, c, cannot be found.

2

1. Write the equation in the form ydy = xdx. Integrating the left side with respect to y and the right side with respect to x yields

2 3

y x 2 3

= + C, or 3y - 2x = c.

23

4. For y n 3/2 multiply both sides of the equation by 3 + 2y to get the separated equation (3+2y)dy = (3x 1)dx. Integration then yields

Section 2.2

9

1,2 3

3y + y = x x + c.

6. We need x n 0 and | y | < 1 for this problem to be

defined. Separating the variables we get (1-y2)"1/2iy =

x 1dx. Integrating each side yields arcsiny = ln|x|+c, so y = sin[ln|x|+c], x n 0 (note that |y | < 1). Also, y = ± 1 satisfy the D.E., since both sides are zero.

10a. Separating the variables we get ydy = (12x)dx, so 2

y2

= x x + c. Setting x = 0 and y = 2 we have 2 = c 2

and thus y2 = 2x 2x2 + or y = \j 2x 2x2 + 4 . The negative square root must be used since y(0) = 2.

10c. Rewriting y(x) as V2(2x)(x+1) , we see that y is

defined for 1 < x < 2, However, since y' does not exist for x = 1 or x = 2, the solution is valid only for the open interval 1 < x < 2.

13. Separate variables by factoring the denominator of the

2x

right side to get ydy = dx. Integration yields

1+x2

22

y /2 = ln(1+x )+c and use of the I.C. gives c = 2. Thus 2 1/2

y = ± [2ln(1+x ) +4] , but we must discard the plus

2

square root because of the I.C. Since 1 + x > 0, the solution is valid for all x.

15. Separating variables and integrating yields 22

y + y = x + c. Setting y = 0 when x = 2 yields c = -4 22

or y + y = x -4. To solve for y complete the square on

the left side by adding 1/4 to both sides. This yields

2 1 2 1 1 2 2 y + y + = x - 4 + or (y + ) = x - 15/4. Taking

4 4 2

the square root of both sides yields

y + = ±Vx2 - 15/4 , where the positive square root must be taken in order to satisfy the I.C. Thus

1 l~2 2

y = - + yx - 15/4 , which is defined for x > 15/4 or

2

x > V15 /2. The possibility that x < -y/15 /2 is discarded due to the I.C.

2 x

17a. Separating variables gives (2y5)dy = (3x e )dx and

10

Section 2.2

2 3 x

integration then gives y 5y = x e + c. Setting x = 0 and y = 1 we have 1 5 = 0 1 + c, or c = 3.

2 3 x

Thus y 5y (x e 3) = 0 and using the quadratic formula then gives

5 ± V 25+4(x3ex 3) 5 [13 3 7

y(x) = ------------------------- = 1 + x3 ex . The

2 2 V 4

negative square root is chosen due to the I.C.

17c. The interval of definition for y must be found

numerically. Approximate values can be found by plotting

13 3 x

y-, (x) = + x and y2 (x) = e and noting the values of x

1 4 2

where the two curves cross.

19a. As above we start with cos3ydy = -sin2xdx and integrate 11

to get sin3y = cos2x + c. Setting y = p/3 when 32

1

x = p/2 (from the I.C.) we find that 0 = - + c or

2

1 1 1 1 2

c = , so that sin3y = cos2x + = cos x (using the 2 3 2 2

appropriate trigonometric identity). To solve for y we

must choose the branch that passes through the point

(p/2,p/3) and thus 3y = p - arcsin(3cos2x), or

p 1 2

y = - arcsin(3cos x).

33

19c. The solution in part a is defined only for

0 < 3cos2x < 1, or V1/3 < cosx < V1/3 . Taking the

indicated square roots and then finding the inverse cosine of each side yields .9553 < x < 2.1863, or Ixp/2| < 0.6155, as the approximate interval.

2 2 3 2

21. We have (3y -6y)dy = (1 + 3x )dx so that y -3y =

x + x3 - 2, once the I.C. are used. From the D.E., the integral curve will have a vertical tangent when 2

3y - 6y = 0, or y = 0,2. For y = 0 we have x3 + x - 2 = 0, which is satisfied for x = 1, which is the only zero of the function w = x3 + x - 2. Likewise, for y = 2, x = -1.

2

23. Separating variables gives y dy = (2+x)dx, so

Section 2.2

11

2

-1 x

-y = 2x + + c. y(0) = 1 yields c = -1 and thus 2

-1 2

y = ----------------- = . This gives

22 x2 2 - 4x - x2

+ 2x - 1 2

dy 8 + 4x

= ------------, so the minimum value is attained at

dx (2-4x-x2) 2

x = -2. Note that the solution is defined for -2 - yf? < x < -2 + \[~6 (by finding the zeros of the denominator) and has vertical asymptotes at the end points of the interval.

2

25. Separating variables and integrating yields 3y + y = sin2x + c. y(0) = -1 gives c = -2 so that 2

y + 3y + (2-sin2x) = 0. The quadratic formula then

gives y = --- + \Jsin2x + 1/4 , which is defined for

-.126 < x < 1.697 (found by solving sin2x = -.25 for x and noting x = 0 is the initial point). Thus we have dy cos2x

= ------------, which yields x = p/4 as the only

dx 1

(sin2x+ )

4

critical point in the above interval. Using the second derivative test or graphing the solution clearly indicates the critical point is a maximum.

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