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  4. The solution of the differential equation y ln y+x y prime=0, where y(1)=e, is

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Q. The solution of the differential equation $y \ln y+x y^{\prime}=0$, where $y(1)=e$, is

Differential Equations

Solution:

$x \frac{ dy }{ dx }+ y (\ln y )=0 \Rightarrow \int \frac{ dx }{ x }+\int \frac{ dy }{ y (\ln y )}= C $
$\ln ( x \ln y )= C . \text { If } x =1 \text { then } y = e $
$\Rightarrow \ln (\ln e )= C \Rightarrow C =0 \Rightarrow( A )]$