Suppose we want to solve the ODE
$$ \frac{dx}{dt} = f(t)x(t)$$ with initial condition $$x(0) = c_{1}+c_{2} \int_{0}^{\infty} g(t)x(t)dt $$ where $f$ and $g$ are continuous function. Then is it possible to find the explicit solution of this differential equation?
I derived the solution but I got it in the implicit form i.e. solution itself depends on $x$.
As
$$ x(t) = e^{\int_0^t f(\tau)d\tau}c_0 $$
after substitution we have
$$ x(0) = c_{1}+c_{2} \int_{0}^{\infty} g(t)e^{\int_0^t f(\tau)d\tau}c_0dt $$
and finally solving for $c_0$
$$ x(0) = c_{1}+c_{2}\phi(c_0) $$