(21 pts) Solve the IVP:

Given the equation: dx + (x/y - sin(y))dy = 0. Find an integrating factor in the form m(y) and then solve the equation.

### Solution

M(x, y) = 1, N(x, y) = x/y - sin(y). M

_{y}= 0, N_{x}= 1/y. The equation is not exact. An integrating factor m could be found from (mM)_{y}= (mN)_{x}. If m only depends on y, the above is reduced tom' M + mM _{y}= mN_{x},or

m'/m = (N _{x}- M_{y}) / M.The latter equation is only solvable if the right hand side does not depend on x. Let's see if that so. (N

_{x}- M_{y})/M = (1/y - 0)/1 = 1/y. Good. m is to be found from:m'/m = 1/y. dm/m = dy/y. m = y. Now multiply the given equation by y:

ydx + (x - y·sin(y))dy = 0. See that it's exact. Indeed, for this equation, M = y, N = x - y·sin(y), so that M

_{y}= 1 = N_{x}. Let's integrate M w.r.t. to x: y(x, y) = xy + C(y). Differentiate both sides w.r.t. to y to obtain N: x + C'(y) = x - y·sin(y), orC'(y) = -y·sin(y). Integration by parts gives C(y) = y·cos(y) - sin(y).

### Answer: y(x, y) = xy + y·cos(y) - sin(y), or xy + y·cos(y) - sin(y) = const.