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AP EAMCET201920 Apr 2019Morning ShiftMathematicsDeterminantsActual

If (f(x), f^ (x) f^ (x) ), are positive functions and (f(0)=1, f^ (0)=2 ), then the solution of the differential equation ( | array ll f(x) & f^ (x) f^ (x) & f^ (x) array |=0 ) is

Options

  1. A(e^ 2 x )
  2. B(2 x+1 )
  3. C( ^2 x+2 x+1 )
  4. D(e^ 4 x )

Correct answer

A. (e^ 2 x )

Step-by-step solution

( aligned & Since, | array ll f(x) & f^ (x) f^ (x) & f^ (x) array |=0 & f(x) f^ (x)- (f^ (x) )^2=0 & f(x) f^ (x)= (f^ (x) )^2 & f^ (x) f^ (x) = f^ (x) f(x) aligned ) On, integrating both sides, we get ( aligned & f^ (x) f^ (x) d x= f^ (x) f(x) d x & _e (f^ (x) )= _e(f(x))+c & f(0)=1 and f^ (0)=2 & c= _e 2 & _e (f^ (x) )= _e(f(x))+ _e 2 & f^ (x)=2 f(x) & f^ (x) f(x) =2 & f^ (x) f(x) d x= 2 d x _e f(x)=2 x+c & f(0)=1 , so c=0 & _e f(x)=2 x f(x)=e^ 2 x aligned ) Hence, option (1) is correct.

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