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  4. The solution of cos y (dy/dx) = ex + sin y + x2esin y is

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Q. The solution of $ cos\,y \frac{dy}{dx} = e^{x + sin \,y} + x^{2}e^{sin \,y}$ is

Differential Equations

Solution:

$cos\, y \frac{dy}{dx} =e^{x} .e^{sin y}+ x^{2} e^{sin y}=e^{sin y }\left(x^{2}+e^{x}\right) $ $\Rightarrow \frac{cos y }{e^{sin y }}\frac{dy}{dx}=\left(x^{2}+e^{x}\right) \Rightarrow \int \frac{cos\, y}{e^{sin\, y }} dy = \int \left(x^{2}+e^{x}\right)dx$
$sin \,y = t$
$cosy \,dy = dt$
$\Rightarrow \int e^{-t} dt = \frac{x^{3}}{3} + e^{x}+C'$
$\Rightarrow \frac{e^{-t}}{-1} = \frac{x^{3}}{3} + e^{x} + C' \Rightarrow e^{x}+e^{- sin\,y} + \frac{x^{3}}{3} = C$