Answer
Not exact differential equation
Work Step by Step
We will use the necessary and sufficient condition for exactness of differential equation according to which
A differential equation \[M(x,y)dx+N(x,y)dy=0\] is called exact differential equation if and only if
\[\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}\]
We have given that
$(y^2+\cos x)dx+2xy^2dy=0$ ___(1)
Here,
$M(x,y)=y^2+\cos x\;\;,\;\;N(x,y)=2xy^2$
$\frac{\partial M}{\partial y} =2y\;\;\;,\;\;\; \frac{\partial N}{\partial x} =2y^2$
$\Rightarrow
\frac{\partial M}{\partial y}\neq\frac{\partial N}{\partial x}$
By necessary and sufficient condition of exactness
That's why (1) is not exact differential equation
