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Question

The solution of the differential equation left parenthesis sin invisible function application x plus cos invisible function application x right parenthesis d y plus left parenthesis cos invisible function application x minus sin invisible function application x right parenthesis d x equals 0 is

  1. e to the power of x end exponent left parenthesis sin invisible function application x plus cos invisible function application x right parenthesis plus c equals 0    
  2. e to the power of y end exponent left parenthesis sin invisible function application x plus cos invisible function application x right parenthesis equals c    
  3. e to the power of y end exponent left parenthesis cos invisible function application x minus sin invisible function application x right parenthesis equals c    
  4. e to the power of x end exponent left parenthesis sin invisible function application x minus cos invisible function application x right parenthesis equals c    

The correct answer is: e to the power of y end exponent left parenthesis sin invisible function application x plus cos invisible function application x right parenthesis equals c


    fraction numerator d y over denominator d x end fraction equals negative fraction numerator cos invisible function application x minus sin invisible function application x over denominator sin invisible function application x plus cos invisible function application x end fraction Þ d y equals negative open parentheses fraction numerator cos invisible function application x minus sin invisible function application x over denominator sin invisible function application x plus cos invisible function application x end fraction close parentheses d x
    On integrating both sides, we get
    Þ y equals negative log invisible function application left parenthesis sin invisible function application x plus cos invisible function application x right parenthesis plus log invisible function application c
    Þ y equals log invisible function application open parentheses fraction numerator c over denominator sin invisible function application x plus cos invisible function application x end fraction close parentheses Þ e to the power of y end exponent left parenthesis sin invisible function application x plus cos invisible function application x right parenthesis equals c.

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