This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Section 1.6 , 4
10 xvv =2V2+1; I :dv = ﬂ; ln(2v2+l) = 4lnx+lnC
2v +1 x 2y2/x2+1= Cx“; x2+2y2 = Cx6 16. The substitution v = x + y+ 1 leads to x_J dv _J2udu (v_u2)
1+J17 1+u 2u—21n(1+u)+C x = 2./x+y+ —21n(1+,/x+y+1)+C
23. v=y'“3; v'—2v/x = —l; p=x'2; y = (x+Cx2)_3 29. The substitution v = sin y yields the homogeneous equation 2xv v’ = 4x2 + v2.
Solution: sinzy = 4x2 — C x
35. F = I063 +y/x)dx = %x4 +ylnx+g(y); Fy = lnx+g’(y) = y2 +lnx = N g'(y) = y2; g(y) = %y3; %x3+;—y2+ylnx = C 36. F = I(l+ye"y)dx = x+exy+g(y); F = xe‘y+g'(y) = 2y+xexy= N y
g’(y) = 2y; g(y) = y’; x+e“’+y2 = C
51. The substitution y’ = p, y" = p p’ = p(dp/dy) in y" = 2 y( y’)3 yields , d 1
pp = 2w3 => J; = IZydy => —; = y2+C. x = Jridy = ly3Cx+D,
p 3 y3+3x+Ay+B = O 59. The substitution x u — 1, y = v — 2 yields the homogeneous equation dv u—v E _ u+v'
The substitution v = pu leads to lnu = JME— = —l[ln(p2+2p—1)—lnC]. (F2 +21) *1) 2
We thus obtain the implicit solution
u2(p2+2p—1)= C
2
u2[3—2—+211)= v2+2uv—u2 = C
u u
(y+2)2+2(x+1)(y+2)—(x+1)2 = C
y2+2xyx2+2x+6y = C.
69. With a = 100 and k = 1/10, Equation (19) in the text is
y = 50[(x/100)9/10— (x/100)‘“‘°].
The equation y’(x) = 0 then yields
(x/100)“‘° = (9/11)”, so it follows that ymix = 50[(9/11)9’2—(9/11)”’2] e 3.68 mi. 71. (a) With a = 100 and k = w/ v0 = 2/ 4 = 1/ 2, the solution given by equation (19) in the textbook is y(x) = 50[(x/100)“2 — (x/100)3/2]. The fact that y(0) = 0 means that
this trajectory goes through the origin where the tree is located. (b) With k = 4/4 = l the solution is y(x) = 50[1 — (x/100)2] and we see that the
swimmer hits the bank at a distance y(0) = 50 north of the tree. (c) With k = 6/4 = 1 the solution is y(x) = 50[(x/100)_“2 — (x/100)5/2]. This
trajectory is asymptotic to the positive x—axis, so we see that the swimmer never reaches
the west bank of the river. ...
View
Full Document
 Spring '07
 TERRELL,R
 Math

Click to edit the document details