(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.
M(x, y) = 1, N(x, y) = x/y - sin(y). My = 0, Nx = 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 to
The latter equation is only solvable if the right hand side does not depend on x. Let's see if that so. (Nx - My)/M = (1/y - 0)/1 = 1/y. Good. m is to be found from:
dm/m = dy/y. m = y. Now multiply the given equation by y:
See that it's exact. Indeed, for this equation, M = y, N = x - y·sin(y), so that My = 1 = Nx. 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), or
Integration by parts gives C(y) = y·cos(y) - sin(y).