1. (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). 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

    m' M + mMy = mNx,

    or

    m'/m = (Nx - My) / M.

    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:

    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 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

    C'(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.