Question

If $y=y\left(x\right)$ is the solution of the equation $e^{\sin y}\cos y\frac{dy}{dx}+e^{\sin y}\cos x=\cos x,$ $y\left(0\right)=0;$ 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

$e^{\sin y}\cos y\frac{dy}{dx}+e^{\sin y}\cos x=\cos x$
Put $e^{\sin y}=t$
$\Rightarrow e^{\sin y}\times \cos y\frac{dy}{dx}=\frac{dt}{dx}$
Then, $\frac{dt}{dx}+t\cos x=\cos x$
I.F. $=e^{\int \cos xdx}=e^{\sin x}$
Solution of differential equation :
$t\cdot e^{\sin x}=\int e^{\sin x}\cdot \cos xdx$
$\Rightarrow e^{\sin y}\cdot e^{\sin x}=e^{\sin x}+C$
At $x=0,y=0$
$\Rightarrow 1=1+C\Rightarrow C=0$
$\therefore \sin y+\sin x=\sin x$
$\Rightarrow y=0$
$\Rightarrow y\left(\frac{\pi }{6}\right)=0,y\left(\frac{\pi }{3}\right)=0,y\left(\frac{\pi }{4}\right)=0$
Required answer is $1+0+0+0=1$