29.10. EXERCISES 575
(f)(y(tan2 xy+1
)+ ycosxy
)dx+
(x(tan2 xy+1
)+ xcosxy+1
)dy = 0,
(0,1)
44. Find the solution curve to the following ODEs which contains the given point.
(a)(2y3 +2
)dx+
(3xy2
)dy = 0,(1,1)
(b)(2y3 +2y+2cos
(x2))
dx+(3xy2 + x
)dy = 0,(1,1)
(c)(2xy2 + y+2xycosx2
)dx+
(2sinx2 +3x2y+2x
)dy = 0,(2,1)
(d) 3y4dx+(
4xy3 + 5y4
x2
)dy = 0, (1,2)
(e)(5x4y+4x3y3
)dx+
(3x5 +5x4y2
)dy = 0,(1,1)
(f)(8x4y6 +3x3
)dx+
(12x5y5 +3xy2
)dy = 0,(−1,2)
45. Explain why every separable ODE can be considered as an exact ODE.
46. Suppose you have a family of level curves f (x,y) = C where C is a constant. Alsosuppose that f is a harmonic function. That is fxx + fyy = 0. Consider the problemof finding another family of level curves such that each of these is perpendicular tothe original level curves f (x,y) = C at any point on both of them. Show that theappropriate equation to solve is 0 = fydx− fxdy. Verify that this is an exact equation.Thus there exists g(x,y) such that the solutions are g(x,y) =C.
M,N Both Affine Linear
47. Find the integral curve for the following differential equation which contains thegiven point. These are also exact so you could use either method.
(a) (2x+ y−3)dx+(x+ y−3)dy = 0,(1,6)
(b) (y− x+2)dx+((x− y)−2)dy = 0,(3,2)
(c) (x+ y−3)dx+(x+3y−7)dy = 0,(2,2)
(d) (2x+ y−8)dx+(x+ y−7)dy = 0,(−2,1)
(e) (x+ y−2)dx+(x+3y−4)dy = 0 = 0,(4,1)
(f) (y−2x+5)dx+(x+ y+2)dy = 0,(1,1)
(g) (y−4x+3)dx+(x−5y+4)dy = 0,(2,1)
48. Find the integral curves for the following differential equation.
(a) (2y− x)dx = (4x+ y−9)dy
(b) (5x+4y−13)dx = (8x+ y−10)dy
(c) (3x−2y+1)dx = (y−4x−3)dy
(d) (4y−4x+4)dx = (8x+ y+11)dy
(e) (2y− x−3)dx = (4x+ y+21)dy
(f) (5y−6x+23)dx = (10x+ y−29)dy
An Assortment of Exercises