- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
14.4 Exercises
- Find a vector-valued function $F,$ continuous on the interval $(0, +\infty),$ such that: $$ \begin{align*} \\ F(x) = xe^xA + \frac{1}{x}\int_1^x F(t)\,dt \end{align*} $$ for all $x > 0,$ where $A$ is a fixed nonzero vector.
- Multiply both sides by $x$ before differentiating.
- Multiplying both sides by $x,$ we get$$xF(x) = x^2e^xA + \int_1^x F(t)\,dt$$Taking the derivative with respect to $x,$ we get $$F(x) + xF'(x) = 2xe^xA + F(x)$$Simplifying terms gives us $$F'(x) = 2e^xA + xe^xA.$$Integrating both sides using the second fundamental theorem with $F(1) = eA,$ we get \begin{align*} F(x) &= eA + \int_1^x 2e^tA + te^tA\,dt \\&= 2e^tA\,\Biggr|_1^x + \int_1^x te^tA\,dt + eA \\&= e^tA + te^tA\,\Biggr|_1^x + eA \\&= e^x(x + 1)A - eA \quad \blacksquare \end{align*}