Maths-
General
Easy

Question

The solution of the differential equation fraction numerator d y over denominator d x end fraction equals fraction numerator x y over denominator x to the power of 2 end exponent plus y to the power of 2 end exponent end fraction is

  1. a y to the power of 2 end exponent equals e to the power of x to the power of 2 end exponent divided by y to the power of 2 end exponent end exponent    
  2. a y equals e to the power of x divided by y end exponent    
  3. y equals e to the power of x to the power of 2 end exponent end exponent plus e to the power of y to the power of 2 end exponent end exponent plus c    
  4. y equals e to the power of x to the power of 2 end exponent end exponent plus y to the power of 2 end exponent plus c    

The correct answer is: a y to the power of 2 end exponent equals e to the power of x to the power of 2 end exponent divided by y to the power of 2 end exponent end exponent


    Given fraction numerator d y over denominator d x end fraction equals fraction numerator x y over denominator x to the power of 2 end exponent plus y to the power of 2 end exponent end fraction. Put y equals v x; fraction numerator d y over denominator d x end fraction equals v plus x. fraction numerator d v over denominator d x end fraction
    therefore v plus x fraction numerator d v over denominator d x end fraction equals fraction numerator left parenthesis x right parenthesis left parenthesis v x right parenthesis over denominator x to the power of 2 end exponent plus v to the power of 2 end exponent x to the power of 2 end exponent end fraction
    Þ v plus x. fraction numerator d v over denominator d x end fraction equals fraction numerator v over denominator 1 plus v to the power of 2 end exponent end fractionÞ x fraction numerator d v over denominator d x end fraction equals fraction numerator negative v to the power of 3 end exponent over denominator 1 plus v to the power of 2 end exponent end fraction
    Þ fraction numerator left parenthesis 1 plus v to the power of 2 end exponent right parenthesis over denominator v to the power of 3 end exponent end fraction d v equals negative fraction numerator d x over denominator x end fractionÞopen parentheses fraction numerator 1 over denominator v to the power of 3 end exponent end fraction plus fraction numerator 1 over denominator v end fraction close parentheses d v equals negative fraction numerator d x over denominator x end fraction
    Integrating both sides, not stretchy integral fraction numerator d v over denominator v to the power of 3 end exponent end fraction plus not stretchy integral fraction numerator d v over denominator v end fraction equals negative not stretchy integral fraction numerator d x over denominator x end fraction
    Þ negative fraction numerator 1 over denominator 2 v to the power of 2 end exponent end fraction plus log invisible function application v equals negative log invisible function application x minus log invisible function application c
    Þ negative fraction numerator x to the power of 2 end exponent over denominator 2 y to the power of 2 end exponent end fraction plus log invisible function application y equals negative log invisible function application cÞlog invisible function application c y equals fraction numerator x to the power of 2 end exponent over denominator 2 y to the power of 2 end exponent end fraction
    Þ c y equals e to the power of x to the power of 2 end exponent divided by 2 y to the power of 2 end exponent end exponent Þ c to the power of 2 end exponent y to the power of 2 end exponent equals e to the power of x to the power of 2 end exponent divided by y to the power of 2 end exponent end exponent
    therefore y to the power of 2 end exponent a equals e to the power of x to the power of 2 end exponent divided by y to the power of 2 end exponent end exponent, where c to the power of 2 end exponent equals a.

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