First slide

Methods of solving first order first degree differential equations

Question

$Solution of differential equation \frac{d y}{d x} + x \sin^{2} y = \sin y \cos y is$

Moderate

A

$\tan y = (x - 1) + C e^{- x}$
B

$\cot y = (x - 1) + C e^{- x}$
C

$\tan y = (x - 1) e^{x} + C$
D

$\cot y = (x - 1) e^{x} + C$

Solution

$\begin{array}{l} \frac{d y}{d x} + x \sin^{2} y = \sin y \cos y \\ \cos e c^{2} y \frac{d y}{d x} + x = \cot y \\ Let - \cot y = v \\ \frac{d v}{d x} + v = x \\ I F = e^{\int 1 d x} = e^{x} \\ s o l : v e^{x} = \int e^{x} x d x \\ ∴ - \cot y \cdot e^{x} = x \int e^{x} d x - \int (1 \int e^{x} d x) d x \\ \Rightarrow \cot y = (x - 1) + C e^{- x} \end{array}$

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