QJEEM10Ma1d58: JEE Main 2010 Mathematics - Class 12 JEEM10Ma1d58 Solution: Solution of the differential equation is :

Question

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Solution of the differential equation

\cos x\,dy = y\left( {\sin x - y} \right)dx,\,\,0 &lt; x &lt;{\pi \over 2}

is :

Accepted Answer

\cos xdy = y\left( {\sin x - y} \right)dx

{{dy} \over {dx}} = y\tan x - {y^2}\,\,\sec \,x

{1 \over {{y^2}}}{{dy} \over {dx}} - {1 \over y}\,\tan \,x = - \sec \,x\,....\left( i \right)

Let

\,\,\,\,{1 \over y} = t \Rightarrow - {1 \over {{y^2}}}{{dy} \over {dx}} = {{dt} \over {dx}}

From equation

(i)

- {{dt} \over {dx}} - \tan x = - \sec x

\Rightarrow {{dt} \over {dx}} + \left( {\tan x} \right)t = \sec x

{\rm I}.F. = {e^{\int {\tan xdx} }} = {\left( e \right)^{\log \left| {\sec x} \right|}}\sec x

Solution

:

\,\,t\left( {{\rm I}.F} \right) = \int {\left( {{\rm I}.F} \right)\sec xdx}

\Rightarrow {1 \over y}\sec x = \tan x + c

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