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Solve the following differential equation dy/dx+ylogy/x-logy =0

Solve the following differential equation
dy/dx+ylogy/x-logy =0

Grade:12th pass

1 Answers

Arun
25763 Points
3 years ago
 
By rearranging the give differential equation we get , dx/dy =1/y -x/(ylogy)
   dx/dy + x/(ylogy) =1/y
    Now, calculate the integrating factor (IF),
    e∫1/(ylogy)=I.F
    Taking, logy=t (say)
     differentiating on both sides we get, dy=ydt
      IF=e∫dt/t=elogt=t, where t=logy
     now, solving in linear differential equation form
    x(t)= ∫t/y dy, since dy=ydt
   xt = ∫t
 xt=t2/2 +c
ie x(logy)=(logy)2/2 =c

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