Problems Determine whether each of the equations in Problems 1 through 8 14. Show that any separable equation is exact. If it is exact, find the solution. 1. (4x + 3) + (6y - 1)y' = 0 M (x) + N(y)y' =0 2. (3x - y) + (x - 3y)y' = 0 3. (3x2 - 4xy + 2) + (6y2 - 2x2 + 3) y' = 0 is also exact. dy ax + by 4. cy - bx In each of Problems 15 and 16, show that the given equation is not dy ax + by exact but becomes exact when multiplied by the given integrating 5. dx bx + cy factor. Then solve the equation. 6. (e" sin y - 3y sin x) + (e cosy + 3 cos x)y' = 0 15. xly3 + x (1 +y?) y' = 0, u(x,y) = 1/xy' 7. (y/ (2x) + 6x) + (In(2x) -2)y' = 0, x>0 16. 2xay + x(1 + y )y'=0, u(x,y) = 1/xy' 3x 3y 8. dy = 0 17. Show that if (N, - M,)/M = Q, where Q is a function of y only, (x2 + 12) 3/2 (x2 + y2)3/2 dx then the differential equation In each of Problems 9 and 10, solve the given initial value problem M + Ny' = 0 and determine at least approximately where the solution is valid. has an integrating factor of the form 9. (2x - y) + (2y - x)y' = 0, y(1) =4 10. (9x7 + y -1) -(4y -x)y'= 0, )(1) =1 H (v) = exp (Q()dy). In each of Problems 11 and 12, find the value of b for which the given In each of Problems 18 through 21, find an integrating factor and equation is exact, and then solve it using that value of b. solve the given equation. 11. (xy' + bx y) + (2x + y)xy' = 0 18. (3x y + 2xy + 13) + (x + y)y' =0 12. (very + 3x) + bxe y' = 0 19. [4(x3/y?) + (3/y)] + [3(x/}?) + 2yly' = 0 13. Assume that equation (6) meets the requirements of Theorem 20. 1 + (x/y - sin ())y' = 0 2.6.1 in a rectangle R and is therefore exact. Show that a possible 21. 1 + (x/y - cos y)y' = 0 function (x, y) is 22. Solve the differential equation 4 (x,y) = M (s, yo) ds + N (x,t) di, ( 6xy + y?) + (2x2 + xy)y' = 0 where (Xo. Yo) is a point in R. Using the integrating factor u (x, y) = 1/ (2xy (4x + y)). Verify that the solution is the same as that obtained in Example 2.6.4 with a different integrating factor