Question

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

• 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. $e^x-e^{\sin y}+\frac{x^3}{3}=c$

Answer 1

Answer: (C)

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 $LHS\Rightarrow \cos ydy=dt$
$\therefore$ $\int \frac{dt}{e^t}=\int (e^x+x^2)dx$
$\Rightarrow$ $-e^{-t}=e^x+\frac{x^3}{3}-c$
$\Rightarrow$ $e^x+e^{-\sin y}+\frac{x^3}{3}=c$