$$\phi(u)=\int_0^{\infty} \lambda e^{-\lambda t}\int_0^{u+ct}f(x)\phi(u+ct-x)dxdt~~~(1)$$ Substituting $s = u + ct$ in the equation 1, $$\phi(u)=\dfrac{1}{c}\int_u^{\infty} \lambda e^{-\lambda (s-u)/c}\int_0^{s}f(x)\phi(s-x)dxds $$ $$ =\dfrac{1}{c} e^{\lambda u/c} \int_u^{\infty} \lambda e^{-\lambda s/c}\int_0^{s}f(x)\phi(s-x)dxds ~~~(2)$$ We can establish an equation for $\phi$, known as an integro-differential equation, by differentiating equation (2) and the resulting equation can be used to derive explicit solutions for $\phi$. Differentiation gives
$$\dfrac{d}{du}\phi(u)=\dfrac{\lambda}{c}\phi(u)- \dfrac{\lambda}{c}\int_0^u f(x)\phi(u-x)dx ~~~(3)$$
How equation (3) has been derived?
By the fundamental theorem of calculus, $$ \Theta(u) = \int_u^\infty \theta(s) d s $$ is a primitive of $-\theta$ when $\theta$ is continuous and the integral exists, so that $\Theta^\prime(u) = - \theta(u)$