Find the function ' f ' which satisfies the equation d f d x =2 f , given that f(0)=e^3

Step-by-step solution

The given differential equation is df dx = 2f . Separating the variables, we have df f = 2 dx . Integrating both sides, df f = 2 dx , which gives |f| = 2x + C . Exponentiating both sides, f(x) = e^ 2x + C = Ae^ 2x , where A = e^C . Using the initial condition f(0) = e^3 , we substitute x = 0 into the equation: e^3 = Ae^ 2(0) = A(1) , so A = e^3 . Substituting the value of A back into the expression for f(x) , we get f(x) = e^3 e^ 2x = e^ 2x + 3 . Answer: e^ 2 x+3