The function `f(x) ` satisfying the equation `f^2 (x) + 4 f'(x) f(x) + (f'(x))^2 = 0`

Options

A. `f(x)+4f'(x).f(x)+[f'(x)]^(2)=0`
B. ` f(x)=-C.e^(2+sqrt(3)x)`
C. ` f(x)= C.e^((sqrt(3)-2)x)`
D. ` f(x) = C.e ^((2-sqrt(3))x)`

Answer

a

Explanation

` {f(x)}^(2) + 4f(x) * f'(x) + {f'(x)}^(2)=0`
Solving for f'(x) , we get
` f'(x) = (-4f(x)pm sqrt(16{f(x)}^(2)-3{f(x)}^(2)))/2`
` - 2 f(x) pm sqrt(3) f(x)`
` :. int (f'(x))/(f(x)) = int (-2 pm sqrt(3))`
` rArr log f(x) = (-2 pm sqrt(3)) x C_(1)`
` :. f(x) = C*e ^(-2 pmsqrt(3)x) " " [ :' C = e^(c_(1))]`