First slide

Methods of solving first order first degree differential equations

Question

$If y = y (x) and \frac{2 + \sin x}{y + 1} \frac{dy}{dx} = - \cos x, y (0) = 1 then y (π / 2) is$

Moderate

A

1/3
B

2/3
C

-1/3
D

1

Solution

$\begin{array}{l} \frac{d y}{y + 1} = \frac{- \cos x}{2 + \sin x} d x \Rightarrow \ln (y + 1) = - \log (2 + \sin x) + \log c \\ \Rightarrow y + 1 = \frac{c}{2 + \sin x}, put x = 0, y = 1 \to c = 4 \\ y (\frac{π}{2}) + 1 = \frac{4}{2 + \sin \frac{π}{2}} = \frac{4}{3} \\ \Rightarrow y (\frac{π}{2}) = \frac{1}{3} \end{array}$

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