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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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Plabita

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