Solution to Review Problems for Midterm II

You should do problems involving the exact equation. Here is one example. Solve (2xy + y + 1) + (3xy + 4xy + 3y) dy dx = 0. Solution: Let M(x, y) = 2xy + y + 1 and N(x, y) = 3xy + 4xy + 3y, We have have My = 6xy 2 + 4y and Nx = 6xy 2 + 4y. So My = Nx and the given equation is exact. Thus there is a function φ(x, y) such that φx = M = 2xy +y+1 and φy = N = 3x y+4xy+3y. Integrating the first equation, we have φ(x, y) = ∫ (2xy+ y+1)dx = xy+xy+x+h(y). Using φy = N = 3x y + 4xy + 3y, we have ∂(x 2y3+xy4+x+h(y)) ∂y = 3xy + 4xy + 3y, 3xy + 4xy + h′(y) = 3xy + 4xy + 3y, h′(y) = 3y and h(y) = ∫ (3y)dy = y + c. Hence φ(x, y) = xy + xy + x + h(y) = xy + xy + x + y and the solution satisfies φ(x, y) = xy + xy + x+ y = c.