Question

The solution of $cosy\frac{dy}{dx}=e^{x+siny}+x^2{e}^{siny}$ is

• A. $e^x-e^{-siny}+\frac{x^3}{3}=C$
• B. $e^{-x}-e^{-siny}+\frac{x^3}{3}=C$
• C. $e^x+e^{-siny}+\frac{x^3}{3}=C$
• D. $e^x-e^{siny}+\frac{x^3}{3}=C$

Answer 1

Answer: (C)

$cosy\frac{dy}{dx}=e^x.e^{siny}+x^2{e}^{siny}=e^{siny}\left(x^2+e^x\right)$ $\Rightarrow \frac{cosy}{e^{siny}}\frac{dy}{dx}=\left(x^2+e^x\right)\Rightarrow \int \frac{cosy}{e^{siny}}dy=\int \left(x^2+e^x\right)dx$
$siny=t$
$cosydy=dt$
$\Rightarrow \int e^{-t}dt=\frac{x^3}{3}+e^x+C^{\prime }$
$\Rightarrow \frac{e^{-t}}{-1}=\frac{x^3}{3}+e^x+C^{\prime }\Rightarrow e^x+e^{-siny}+\frac{x^3}{3}=C$