Let f: R R be a differentiable function satisfying f^ (x)+2 f(x)= , where R . If f(0)=5 and f ( 1 2 _ e 3 )=7 , then the value of + _ x f(x) is equal to ________.

Step-by-step solution

The given differential equation is f^ (x)+2 f(x)= , which is a linear differential equation of the first order. The integrating factor (I.F.) is e^ 2 d x = e^ 2x . Multiplying the equation by the I.F. and integrating, we get: f(x) e^ 2x = e^ 2x d x f(x) e^ 2x = 2 e^ 2x + C f(x) = 2 + C e^ -2x Using the condition f(0) = 5 : 2 + C = 5 --- (1) Using the condition f ( 1 2 _ e 3 ) = 7 : 2 + C e^ -2 ( 1 2 _ e 3 ) = 7 2 + C e^ - _ e 3 = 7 2 + C 3 = 7 --- (2) Subtracting equation (2) from equation (1): C - C 3 = 5 - 7 2C 3