# Integral Equation Solution

I'm looking into solving the following integral equation:

$$y(x)=1+\int^{x}_{0}(\tanh s)y(s)ds$$

How can I go about turning this into a differential equation? i.e. of the form

$$y'(x)=f(y)$$ for some function $f$ so we can then apply $y(0)=1$ to deduce a solution by standard techniques for Differential Equations.

-
differentiate both sides using the fundamental theorem of calculus: $$F(x)=\int_a^xf(t)dt\Rightarrow F'(x)=f(x)$$ $$y'=y\tanh x$$ integrate: $$\log y=\log\cosh x+C$$ $$y=C\cosh x$$ initial conditions: $$y(0)=1=C$$
Differentiate both sides with respect to $x$, using the Fundamental Theorem of Calculus.