A function f satisfies the condition `f(x)=f'(x)+f''(x)+f''(x)+…,` where f(x) is a differentiable function indefinitely and dash denotes the order the derivative. If f(0) = 1, then f(x) is

Options

A. `e^(x//2)`
B. `e^(x)`
C. `e^(2x)`
D. `e^(4x)`

Explanation

`"Given "f=f'+f''+f'''+...oo`
`"or "f'=f''+f'''+f'''+...oo`
`"or "f-f'=f'`
`"or "f=2f'`
`"Hence, "(f')/(f)=1//2 or int(f')/(f)dx=int(1)/(2)dx`
`"or "logf(x)=x//2+c`
`"or "f(x)=e^(x//2+c)`
`"Also, "f(0)=1 or c = 0 or f(x) =e^(x//2)`