I have the following Voltera equation $$ u(x) = e^x + \int_{0}^{x}u(t)dt. $$ To convert it on an ODE, I differentiate and find
$u'(x) = e^x + u(x)$ where $u(0) = 1$ but I see the solution is $$ u'(x) = e^x + u(x) -1$$ Im not finding where the 1 is coming from
I believe the lecture notes simply have an error. I’m using this answer to explain why I think so.
First, if we differentiate both sides of the Volterra equation, then we get $u’(x) = e^x + u(x)$ by the Fundamental Theorem of Calculus. So the $-1$ term is just not there.
Second, as a check, suppose we solve the initial value problem as given in the notes, $u’(x) = e^x + u(x) - 1$, $u(0) = 1$. Then we obtain $u(x) = x e^x + 1$. Inserting this into the right-hand side of Volterra equation gives $x e^x + x + 1$, which differs from $u(x)$. In short, the solution of the initial value problem $u’(x) = e^x + u(x) - 1$, $u(0) = 1$ is not a solution to the Volterra equation.
But the solution of the initial value problem $u’(x) = e^x + u(x)$, $u(0) = 1$, which is $u(x) = x e^x + e^x$, does solve the Volterra equation, as you can check.