Let $y = y(x)$ be the solution of the differential equation $secx\frac{dy}{dx} - 2y = 2 + 3sinx,x \i

Question

Let $y = y(x)$ be the solution of the differential equation $secx\frac{dy}{dx} - 2y = 2 + 3sinx,x \in \left( - \frac{\pi}{2},\frac{\pi}{2} \right)$, $y(0) = - \frac{7}{4}$. Then $y\left( \frac{\pi}{6} \right)$ is equal to:

Accepted Answer

$\frac{dy}{dx} - 2ycosx = 2cosx + 3sinx \cdot cosx$

I.F. $= e^{- 2sinx}$

$${e^{- 2sinx} \cdot y = \int e^{- 2sinx}(3sinxcosx + 2cosx)dx }{y.e^{- 2sinx} = e^{- 2sinx}\left( - \frac{3}{2}sinx - \frac{7}{4} \right) + C }{\Rightarrow y = - \frac{3}{2}sinx - \frac{7}{4} + C.e^{2sinx} }{\because y(0) = - \frac{7}{4} \Rightarrow C = 0 }{y\left( \frac{\pi}{6} \right) = \frac{- 3}{2} \cdot \frac{1}{2} - \frac{7}{4} = \frac{- 5}{2}}$$

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