If y = y(x) is the solution of the equation e ^ y y(dy)/(dx) + e ^...

$⇒ {e}^{\sin y}\cos y\dfrac{dy}{dx} = \dfrac{dt}{dx}$

$⇒ D . E$ is $\dfrac{dt}{dx} + tcosx = \cos x$

$I . F . = {e}^{∫ cosxdx} = {e}^{\sin x}$

$⇒$ solution is $t . {e}^{\sin x} = ∫ \cos x{e}^{\sin x}$

$⇒ {e}^{\sin y}{e}^{\sin x} = {e}^{\sin x} + c$

$∵ x = 0, y = 0 ⇒ c = 0$

$⇒ {e}^{\sin y} = 1$

$⇒ y = 0$

$⇒ 1 + y\left(\dfrac{\pi}{6}\right) + \dfrac{\sqrt{3}}{2}y\left(\dfrac{\pi}{3}\right) + \dfrac{1}{\sqrt{2}}y\left(\dfrac{\pi}{4}\right) = 1$