KCET2013MathematicsDifferential Equations
If f(x)=f^ (x)+f^ (x)+f^ (x)+ and f(0)=1 , then f(x) is equal to
Options
- Ae^ x / 2
- Be^ x
- Ce^ 2 x
- De^ 4 x
Correct answer
A. e^ x / 2
Step-by-step solution
Given, f(x)=f^ (x)+f^ (x)+f^ (x)+ If f(x)=e^ x / 2 Then, f^ (x)= 1 2 e^ x / 2 , f^ (x)= 1 2² e^ x / 2 , aligned f^ (x) &= 1 2³ e^ x / 2 so on f(x) &=f^ (x)+f^ (x)+f^ (x)+ &= 1 2 e^ x / 2 + 1 2² e^ x / 2 + 1 2³ e^ x / 2 aligned =e^ x / 2 ( 1 2 + 1 2² + 1 2³ + )=e^ x / 2 1 2 1- 1 2 =e^ x / 2 ( 1 2 1 2 )=e^ x / 2 1 f(x)=e^ x / 2 and f(0)=e⁰=1