Question

Q\Determine the following ODEs are exact or not? Why?
((y)dx+(x)dy/(x+y)^2)+dy=0

Answer 1

$\displaystyle y\mathrm{d}x + \frac{x\mathrm{d}y}{(x+y)^2} + \mathrm{d}y = 0\\ \textsf{Multiplying through by}\, (x + y)^2\\ y(x + y)^2\mathrm{d}x + x\mathrm{d}y + (x + y)^2\mathrm{d}y = 0\\ y(x + y)^2\mathrm{d}x + (x + (x + y)^2)\mathrm{d}y = 0\\ M = y(x + y)^2\\ \begin{aligned} M_y = \frac{\partial M}{\partial y} &= (x + y)^2 + 2y(x + y)\\ &= (x + y)(x + y + 2y)\\ &= (x + 3y)(x + y) \end{aligned}\\ N = x + (x + y)^2\\ N_x = \frac{\partial N}{\partial x} = 1 + 2(x + y)\\ \textsf{Since}\, N_x \neq M_y,\, \textsf{the differential equation}\\ \textsf{is not exact.}$