esiny cos y \(\frac{dy}{dx}\) + esiny cos x = cos x
esiny = t
esiny. cos y \(\frac{dy}{dx}\) = \(\frac{dt}{dx}\)
\(\therefore\) \(\frac{dt}{dx}\) + cos x t = cos x
\(\therefore\) I.F. = \(e^{\int cosxdx}\) = esinx
\(\therefore\) t x I.F. = \(\int\)(I.F.) \(\times\) α dx
⇒ t.esinx = \(\int e^{sinx}.cos x dx\)
= esinx + c
⇒ t = 1 + c e-sinx
⇒ esiny = 1 + c e-sinx
y(0) = 0
⇒ c + 1 = eo = 1
⇒ c = 0
\(\therefore\) esiny = 1 is solution of given differential equation.
⇒ sin y = ln 1 = 0
⇒ y = sin-10 = 0
\(\therefore\) y = 0 is solution of given differential equation.
\(\therefore\) y(π/3) = y(π/4) = y(π/6) = 0
\(\therefore\) 1 + y(π/6) + \(\frac{\sqrt3}2y(\pi/3)\) + \(\frac1{\sqrt2}y(π/4)\) = 1 + 0
= 1