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If $x\frac{dy}{dx}=y(\log y-\log x+1),$ then the solution of the equation is

Option 1)
$x\log \left ( \frac{y}{x} \right )=cy\;$

Option 2)
$\; y\log \left ( \frac{x}{y} \right )=cx\;$

Option 3)
$\; \log \left ( \frac{x}{y} \right )=cy\;$

Option 4)
$\; \log \left ( \frac{y}{x} \right )=cx\; \;$

Answers (1)

best_answer

As we learnt in

Homogeneous Differential Equation -

Put

$\frac{y}{x}=v$

$\frac{dy}{dx}=v+\frac{xdv}{dx}$

-

$x.\frac{dy}{dx}= y(\log y- \log x +1)$

$\Rightarrow \frac{dy}{dx}= \frac{y}{x}(\log \frac{y}{x}+1)$

Put $\frac{y}{x}= v$

$v+x\frac{dv}{dx}= v+ v\log v$

$\int \frac{dv}{v \log v}= \int \frac{dx}{x}$

$\Rightarrow \log (\log v)= \log x+c= \log \left | cx \right |$

$\Rightarrow \log v= cx$

$\Rightarrow \log \left(\frac{y}{x} \right )= cx$

Option 1)

$x\log \left ( \frac{y}{x} \right )=cy\;$

Incorrect option

Option 2)

$\; y\log \left ( \frac{x}{y} \right )=cx\;$

Incorrect option

Option 3)

$\; \log \left ( \frac{x}{y} \right )=cy\;$

Incorrect option

Option 4)

$\; \log \left ( \frac{y}{x} \right )=cx\; \;$

Correct option

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