First slide

Methods of solving first order first degree differential equations

Question

$The solution of the D.E [x coty + \log \cos x] d y + (logsiny-ytanx) d x = 0 is$

Moderate

A

${(sinx)}^{y} {(cosy)}^{x} =C$
B

${(siny)}^{x} {(cosx)}^{y} =C$
C

${(sinx)}^{x} {(cosy)}^{y} =C$
D

$tanxtany = C$

Solution

$[x coty + \log \cos x] d y + (logsiny-ytanx) d x = 0$
$[x coty + \log \cos x] \frac{d y}{d x} + (logsiny-ytanx) = 0$
$x c o t y \frac{d y}{d x} + \log \cos x \frac{d y}{d x} + \log \sin y - y \tan x = 0$
$x c o t y \frac{d y}{d x} + \log \sin y - y \tan x + \log \cos x \frac{d y}{d x} = 0$
$\begin{array}{l} \int d (xlogsiny) + \int d (ylogcos x) = C \\ \to xlog \sin y + ylog \cos x = C \\ \to (\sin y)^{x} (\cos x)^{y} = C \end{array}$

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