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    Maths-
    General
    Easy

    Question

    The solution of the differential equation square root of a plus x end root fraction numerator d y over denominator d x end fraction plus x y equals 0 is

    1. y equals A e to the power of 2 divided by 3 left parenthesis 2 a minus x right parenthesis square root of x plus a end root end exponent    
    2. y equals A e to the power of negative 2 divided by 3 left parenthesis a minus x right parenthesis square root of x plus a end root end exponent    
    3. y equals A e to the power of 2 divided by 3 left parenthesis 2 a plus x right parenthesis square root of x plus a end root end exponent    
    4. y equals A e to the power of negative 2 divided by 3 left parenthesis 2 a minus x right parenthesis square root of x plus a end root end exponent    

    hintHint:

    Use the separation of variables technique to solve the equation.

    The correct answer is: y equals A e to the power of 2 divided by 3 left parenthesis 2 a minus x right parenthesis square root of x plus a end root end exponent

      square root of a plus x end root space fraction numerator d y over denominator d x end fraction space plus space x y space equals space 0 square root of a plus x end root space fraction numerator d y over denominator d x end fraction space equals space minus x y square root of a plus x end root space d y space equals space minus x y space d x fraction numerator d y over denominator y end fraction space equals space fraction numerator negative x over denominator square root of a plus x end root space end fraction space d x I n t e g r a t i n g space b o t h space t h e space s i d e s comma integral fraction numerator d y over denominator y end fraction space equals space integral fraction numerator negative x over denominator square root of a plus x end root space end fraction space d x log y space equals space integral fraction numerator negative open parentheses x plus a minus a close parentheses over denominator square root of a plus x end root space end fraction space d x log y space equals negative 1 space open parentheses integral fraction numerator open parentheses x plus a close parentheses over denominator square root of a plus x end root space end fraction space d x space minus space integral fraction numerator a over denominator square root of a plus x end root space end fraction space d x close parentheses log y space equals space minus 1 open parentheses integral square root of x plus a end root space d x space minus a space integral fraction numerator 1 over denominator square root of a plus x end root space end fraction space d x close parentheses log y space equals space minus 1 open parentheses 2 over 3 open parentheses x plus a close parentheses to the power of 3 over 2 end exponent space minus 2 a square root of x plus a end root close parentheses space plus space log A space space space space space space space space space space space space space space space open parentheses w h e r e space A space i s space a n space a r b i t r a r y space c o n s tan t close parentheses log y space equals space space 2 a square root of x plus a end root space minus space 2 over 3 open parentheses x plus a close parentheses to the power of 3 over 2 end exponent space plus space log A log y space equals space 2 square root of x plus a end root open parentheses a space minus space fraction numerator open parentheses x plus a close parentheses over denominator 3 end fraction close parentheses space plus space log A log y space equals space 2 square root of x plus a end root open parentheses fraction numerator 3 a minus open parentheses x plus a close parentheses over denominator 3 end fraction close parentheses space plus space log A log y space equals space 2 over 3 square root of x plus a end root open parentheses 3 a minus x minus a close parentheses space plus space log A log y space equals 2 over 3 square root of x plus a end root open parentheses 2 a minus x close parentheses space plus space log A y space equals space e to the power of 2 over 3 open parentheses 2 a minus x close parentheses square root of x plus a end root end exponent space space plus space e to the power of log A end exponent therefore y space equals space A e to the power of 2 over 3 open parentheses 2 a minus x close parentheses square root of x plus a end root end exponent

      A differential equation is an equation which contains one or more terms and the derivatives of one variable (i.e., dependent variable) with respect to the other variable (i.e., independent variable) dy/dx = f(x)

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