Exact Equations:

1. Determine whether each of the equations is exact. If it is exact, find the solution.

(a) \( \displaystyle{(2xy^2+2y) \, dx + (2x^2y+2x) \, dy = 0}\)    solution

(b) \( \displaystyle{\left(1+\ln x + \frac{y}{x}\right) -(1-\ln x)\, y'=0}\)    solution

(c) \( \displaystyle{ (\sin y - y \sin x)\, dx + (\cos x + x \cos y - y) \, dy = 0 }\)    solution

(d) \( \displaystyle{ (e^x \sin y - 2 y \sin x) + (e^x\cos y + 2 \cos x)\, \frac{dy}{dx} = 0 }\)   solution

2. Find the value \(k\) so that the following equation is exact, then find the general solution.    solution

\( \displaystyle{ (5x+ky)\,dx + (4x-3y)\, dy = 0 }\)