I need to learn how to solve differential equations using either the Exact Equation Approach and or the Special Integrating Factor methods. Below is a differential Equation to solve.
$(2xy^2 + \cos x) \text{d}x + (2x^2 y + \sin y)\text{d}y = 0$
I would appreciate it if you would include comments to explain steps taken. Thanks in advance
Following your example I did the following
Given $$ (2x + y).dx + ( x - 2y).dy = 0$$
a) $$ M(x,y)=2x + y, N(x,y)= x - 2y $$
b) check if the d.e is exact.
$\frac{\partial M}{\partial y}=\frac{\partial}{\partial y}\left(2x + y\right)= 1 =\frac{\partial}{\partial x}\left(x - 2y\right)=\frac{dN}{\partial x}$.
c) $$ f\left(x,y\right)=\int M(x,y)\text{d}x =\int(2x + y)\text{d}x=x^{2} + xy + g(y).$$
d) To find $g\left(y\right)$
$f_{y}\left(x,y\right)=\frac{\partial}{dy}\left(x^{2}+ xy + g(y)\right)=0 + x + g'\left(y\right).$
e) Upon comparing with $N\left(x,y\right)$, I find $g'\left(y\right)= - 2y$ which implies that $g\left(y\right)=- y^{2}+K$
Therefore, the general solution is $ x^2 + xy - y^2 =C$.