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  4. The solution of cos y (d y/d x)=ex+ sin y+x2 e sin y is f(x)+e- sin y=C(C is arbitrary real constant) where f(x) is equal to

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Q. The solution of $\cos y \frac{d y}{d x}=e^{x+\sin y}+x^2 e^{\sin y}$ is $f(x)+e^{-\sin y}=C(C$ is arbitrary real constant) where $f(x)$ is equal to

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Solution:

$-e^{-\sin y} \cos y \frac{d y}{d x}=-\left[e^x+x^2\right] $
$\Rightarrow d\left(e^{-\sin y}\right)+\left(e^x+x^2\right) d x=0$
$ \Rightarrow e^{-\sin y}+e^x+\frac{x^3}{3}=C$