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. Show that \(\mu(x,y)=xy^2\) is an integrating factor for \((2y-6x)\,dx+(3x-4x^2y^{-1})\, dy =0\), and use the inegrating factor to solve the diff eqn.   solution

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

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

4. Find an Integrating Factor (to make the equation exact), then use it to solve the differential equation.

(a) \( \displaystyle{(3x^2y+2xy+y^3) \, dx + (x^2+y^2) \, dy = 0}\)    solution

(b) \( \displaystyle{y \, dx + (2xy-e^{-2y}) \, dy = 0}\)     solution

(c) \( \displaystyle{e^{x+2y} \, dx + \left(2+\frac{2}{y}\right)e^{x+2y} \, dy = 0}\)    solution

(d) \( \displaystyle{(2xy^3-2x^3y^3-4xy^2+2x)\, dx + (3x^2y^2+4y)\, dy =0}\)   solution