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Mathematics
The solution of cos y(dy/dx)=ex+ sin y+x2e sin 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
KEAM
KEAM 2009
Differential Equations
A
$ {{e}^{x}}-{{e}^{-\sin y}}+\frac{{{x}^{3}}}{3}=c $
B
$ {{e}^{-x}}-{{e}^{-\sin y}}+\frac{{{x}^{3}}}{3}=c $
C
$ {{e}^{x}}+{{e}^{-\sin y}}+\frac{{{x}^{3}}}{3}=c $
D
$ {{e}^{x}}-{{e}^{\sin y}}-\frac{{{x}^{3}}}{3}=c $
E
$ {{e}^{x}}-{{e}^{\sin y}}+\frac{{{x}^{3}}}{3}=c $
Solution:
Given, $ \cos y\frac{dy}{dx}={{e}^{x+\sin y}}+{{x}^{2}}{{e}^{\sin y}} $
$ \Rightarrow $ $ \cos y\frac{dy}{dx}={{e}^{\sin y}}({{e}^{x}}+{{x}^{2}})dx $
$ \Rightarrow $ $ \int{\frac{\cos y}{{{e}^{\sin y}}}dy}=\int{({{e}^{x}}+{{x}^{2}})dx} $
Put $ \sin y=t $ in $ LHS\Rightarrow \cos ydy=dt $
$ \therefore $ $ \int{\frac{dt}{{{e}^{t}}}}=\int{({{e}^{x}}+{{x}^{2}})}dx $
$ \Rightarrow $ $ -{{e}^{-t}}={{e}^{x}}+\frac{{{x}^{3}}}{3}-c $
$ \Rightarrow $ $ {{e}^{x}}+{{e}^{-\sin y}}+\frac{{{x}^{3}}}{3}=c $