JEE PYQ: Differential Equation Question 14
Question 14 - 2021 (26 Feb Shift 1)
If $y = y(x)$ is the solution of the equation $e^{\sin y}\cos y\frac{dy}{dx} + e^{\sin y}\cos x = \cos x$, $y(0) = 0$; 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 ______.
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Answer: 1
Solution
Put $e^{\sin y} = t$: $\frac{dt}{dx} + t\cos x = \cos x$. I.F. $= e^{\sin x}$. Solution: $te^{\sin x} = e^{\sin x} + c$, i.e. $e^{\sin x + \sin y} = e^{\sin x} + c$. At $(0,0)$: $c = 0$. So $\sin x + \sin y = \sin x$, giving $\sin y = 0$, $y = 0$ for all $x$. Hence the expression $= 1 + 0 + 0 + 0 = 1$.
BETA
