Find constants so that e^x satisfies a given integral equation

by RoRi

November 13, 2015

Find values for the constants $a$ and $b$ such that the following equation is true:

$e^x = b + \int_a^x e^t \, dt.$

To find values of the constants we evaluate the integral,

$\int_a^x e^t \, dt = e^t \Big \rvert_a^x = e^x - e^a.$

Therefore, we need to find constants such that

$\begin{align*} &&e^x &= b + \int_a^x e^t \, dt \\ \implies &&e^x &= b + e^x - e^a \\ \implies && b &= e^a. \end{align*}$

Therefore, we can choose any value for $a$ and then set $b = e^a$ and this pair will make the equation true for all $x$ .

Stumbling Robot