Question

If $y=y(x)$ is the solution of the equaiton $e^{\text{sin y}}\cos y\frac{dy}{dx}+e^{\text{sin y}}\cos x=\cos x,y(0)=0$; then $1+y\left(\frac{\pi }{6}\right)+\frac{\sqrt{3}}{2}y\left(\frac{\pi }{3}\right)+\frac{1}{\sqrt{2}}y\left(\frac{\pi }{4}\right)$ is equal to

Answer 1

Answer: 1

Put $e^{\text{siny }}=t$
$\Rightarrow e^{\sin y}\cos y\frac{dy}{dx}=\frac{dt}{dx}$
$\Rightarrow D.E$ is $\frac{dt}{dx}+t\cos x=\cos x$
$I.F.=e^{\int \cos xdx}=e^{\sin x}$
$\Rightarrow$ solution is $t.e^{\sin x}=\int \cos x{e}^{\sin x}$
$\Rightarrow e^{\sin y}{e}^{\sin x}=e^{\sin x}+c$
$\because x=0,y=0\Rightarrow c=0$
$\Rightarrow e^{\sin y}=1$
$\Rightarrow y=0$
$\Rightarrow 1+y\left(\frac{\pi }{6}\right)+\frac{\sqrt{3}}{2}y\left(\frac{\pi }{3}\right)+\frac{1}{\sqrt{2}}y\left(\frac{\pi }{4}\right)=1$