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Solution of diffrential equation dy/dx = \frac{y}{2\sqrt{xy}+x}  will be 

  • Option 1)

    \sqrt {x/y } - ln |y| = c

  • Option 2)

    \sqrt {x/y } + ln |y| = c

  • Option 3)

    \sqrt {y/x } - ln |y| = c

  • Option 4)

    \sqrt {y/x } + ln |y| = c

 

Answers (1)

best_answer

AS we have learned

Homogeneous Differential Equation -

Put

\frac{y}{x}=v

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

-

 

 \frac{dy}{dx}= \frac{y}{2\sqrt {xy }+x} \Rightarrow \frac{dx}{dy} = \frac{2\sqrt {xy}+x}{y }

Let x/y = v \Rightarrow x = vy \Rightarrow dx\dy = v+y dv/dy

\Rightarrow x+y \frac{dx}{dy} = \frac{2\sqrt{v } y+vy}{y } \Rightarrow v+y dx/dy = 2\sqrt v+v

\Rightarrow y \frac{dv}{dy} = 2\sqrt v\Rightarrow \frac{dv }{ 2\sqrt v }=\frac{dy}{y}\Rightarrow \int dx/2\sqrt v - \int dy/y= c

\Rightarrow \sqrt v - ln |y|= c

\Rightarrow \sqrt {x/y} - ln |y|= c

 

 

 

 


Option 1)

\sqrt {x/y } - ln |y| = c

Option 2)

\sqrt {x/y } + ln |y| = c

Option 3)

\sqrt {y/x } - ln |y| = c

Option 4)

\sqrt {y/x } + ln |y| = c

Posted by

prateek

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