If f(x)=f^ (x) and f(1)=2 , then f(3) is

Step-by-step solution

Given the differential equation f(x) = f'(x) , we can write it as df dx = f(x) . Separating the variables, we have df f = dx . Integrating both sides, we get |f(x)| = x + C , which implies f(x) = Ae^x for some constant A . Using the initial condition f(1) = 2 , we substitute x=1 : 2 = Ae^1 , which gives A = 2 e . Thus, the function is f(x) = ( 2 e )e^x = 2e^ x-1 . To find f(3) , we substitute x=3 : f(3) = 2e³⁻¹ = 2e^2 . Answer: 2 e^2