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AP EAMCET201920 Apr 2019Evening ShiftMathematicsDifferential EquationsActual

The general solution of the differential equation y y^ =x [ y^2 x^2 + ( y^2 x^2 ) ^ ( y^2 x^2 ) ] , where is an arbitrary function, is

Options

  1. Ax ( y^2 x^2 )=c y
  2. Bx^2 ( y^2 x^2 )=c
  3. Cx^2 ( y^2 x^2 )=c y^2
  4. D( y^2 x^2 )=c x^2

Correct answer

D. ( y^2 x^2 )=c x^2

Step-by-step solution

Given, differential equation y y^ =x [ y^2 x^2 + ( y^2 x^2 ) ^ ( y^2 x^2 ) ], where is an arbitrary function. y x d y d x = y^2 x^2 + ( y x )^2 ^ ( y x )^2 Let y=v x , where v is function of x . d y d x =v+x d v d x array ll so & v (v+x d v d x )=v^2+ (v^2 ) ^ (v^2 ) & v x d v d x = (v^2 ) ^ (v^2 ) & v ^ (v^2 ) (v^2 ) d v= d x x array aligned & 1 2 _e (v^2 )= _e x+ _e( c )= _e( c x) & (v^2 )=c x^2 ( y^2 x^2 )=c x^2 aligned Hence, option (d) is correct.

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