First slide

Introduction to Differential equations

Question

$If y - \cos x \cdot \frac{d y}{d x} = y^{2} (1 - \sin x) \cos x, y (0) = 1 then y (\frac{π}{3})$

Moderate

A

1
B

2
C

$\frac{1}{2}$
D

$\sqrt{3}$

Solution

$\begin{array}{l} Dividing the given DE by y^{2} \cos x \\ \frac{- 1}{y^{2}} \cdot \frac{d y}{d x} + \sec x \cdot \frac{1}{y} = 1 - \sin x \\ \Rightarrow \frac{d}{d x} (\frac{1}{y}) + \sec x \cdot \frac{1}{y} = 1 - \sin x \\ I . F = \int_{e} \sec x \cdot d x = \sec x + \tan x \\ Solution if \frac{1}{y} (\sec x + \tan x) = \int \frac{(1 - \sin x) (1 + \sin x)}{\cos x} = \sin x + c \\ x = 0, y = 1, \Rightarrow c = 1 ∴ y = \sec x, \Rightarrow y (\frac{π}{3}) = 2 \end{array}$

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