The solution of ( x + tan y ) ? d y = sin 2 y ? d x is:

Question

The solution of $(x + \tan y) d y = \sin 2 y d x$ is:

Smart Achievers · Accepted Answer

(1)

Ans. $x = \tan y + c \sqrt{\tan y}$

Explanation:

$(x + \tan y) d y = \sin 2 y d x$

$\frac{d x}{d y} = \frac{\sin 2 y}{x + \tan y}$

$\frac{d y}{d x} = \frac{x + \tan y}{\sin 2 y}$

$\frac{d x}{d y} - (\frac{1}{\sin 2 y}) x = \frac{\tan y}{\sin 2 y}$

$I . F . = e^{- \int \frac{1}{\sin 2 y} d y}$

$= e^{- \int c o s e c 2 y d y} = e^{- \frac{1}{2} \log (\tan y)}$

$= \frac{1}{\sqrt{\tan y}}$

The general solution is:

$x \cdot \frac{1}{\sqrt{\tan y}} = \int \frac{1}{\sqrt{\tan y}} \cdot \frac{\tan y}{\sin 2 y} d x$

$\frac{x}{\sqrt{\tan y}} = \frac{1}{2} \int \frac{\sec^{2} y}{\sqrt{\tan y}} d y$

$\frac{x}{\sqrt{\tan y}} = \frac{1}{2} \sqrt{\tan y} + c$

$x = \tan y + c \sqrt{\tan y}$

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