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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.

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2 Answers 2

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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: $$ y(0)=1=C $$

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Differentiate both sides with respect to $x$, using the Fundamental Theorem of Calculus.

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