574 CHAPTER 29. FIRST ORDER SCALAR ODE
(c) y′ = 14x2+yx
(x2 +4xy+ y2
),(−1,1)
(d) y′ =− 13x2−xy
(x2−3xy+ y2
),(1,1)
(e) y′ = 1x(y+5x)
(x2 +5xy+ y2
),(−1,−1)
(f) y′ = 1x(3y+2x)
(x2 +2xy+3y2
),(−2,3)
(g) y′ = 1x(4y−x)
(x2− xy+4y2
),(3,−2)
36. Find the solution curve to the following ODEs which contains the given point.
(a) y′ = 1x2
(x2 + y2 + xy
),(1,1)
(b) y′ = 1x2
(4x2 + y2 + xy
),(2,0)
(c) y′ = 1x2
(x2 +9y2 + xy
),(3,1)
(d) y′ = 1x2
(4x2 +2y2 + xy
),(−1,1)
37. Find the solution curve to the following ODEs which contains the given point.
(a) −(x+ y)dx+(x+2y)dy = 0,(1,1)
(b) (x− y)dx+(x+3y)dy = 0,(2,1)
(c) (4x+ y)dx+(x+2y)dy = 0,(−1,2)
(d) −(3x+ y)dx+(x− y)dy = 0,(3,2)
(e) (3x−4y)dx+(4x− 4
3 y)
dy = 0, (3,1)
(f) (−y)dx+(4y− x)dy = 0, (0,2)
(g)(−2x− 31
4 y)
dx+(x− 9
4 y)
dy = 0, (−1,2) .
38. Find all solutions to y′+ sin( y
x
)= 1. Hint: You might need to leave the answer in
terms of integrals.
39. Solve: x2dy+(4x2− xy+5y2
)dx = 0, y(3) =−1.
40. Solve: x2dy+(7x2− xy+4y2
)dx = 0, y(2) =−1.
41. Solve: x2dy+(6x2− xy+3y2
)dx = 0, y(−1) = 1.
42. Solve:(x3−7x2y−5y3
)dx+
(7x3 +5xy2
)dy = 0, y(3) =−2.
Exact Equations and Integrating Factor
43. Find the solution curve to the following ODEs which contain the given point. Firstverify that the equation is exact.
(a) (2xy+1)dx+ x2dy = 0,(1,1)
(b) (2xsiny+1)dx+(x2 cosy
)dy = 0,
(1, π
2
)(c) (2xsiny− sinx)dx+
((cosy)x2 +1
)dy = 0,(0,0)
(d)(
yxy+1
)dx+ 1
xy+1 (x+ xy+1)dy = 0,(1,1)
(e)(y2 cosxy2 +1
)dx+
(2xycosxy2 +1
)dy = 0,(1,0)