Let f be a twice differentiable function such that f(x)= ₀^ x (t-x)dt- ₀^ x f(t) t ,dt , x (- 2 , 2 ) . Then f'' ( 6 )+12f' (- 6 )+f ( 6 ) is equal to ______

Step-by-step solution

Given the function: f(x) = ₀^ x (t-x) dt - ₀^ x f(t) t , dt First, we simplify the first integral. Let u = x - t , then du = -dt . When t = 0 , u = x ; when t = x , u = 0 . ₀^ x (t-x) dt = _ x ⁰ (-u) (-du) = _ x ⁰ u , du = - ₀^ x u , du = -[ ( u)]₀^ x = - ( x) = ( x) So, the equation becomes: f(x) = ( x) - ₀^ x f(t) t , dt Differentiating both sides with respect to x using the Leibniz rule: f'(x) = - x x - f(x) x f'(x) + f(x) x = - x This is a linear first-order differential equation. The integrating factor (IF) is