Help with numerical simulation of a backwards ODE.

Question

This is a toy problem of a much more complicated ODE system. Suppose I have the following ODE $$\frac{dx}{ds} = kx-c, \quad x(0) = 1$$ Assume that $s<t$ and that $x(s)$ is the value of some multivariate function say $x(s) = \Psi(1,0,t-s,t)$. Now the text says solving the equation for each value of $t$ gives us $S(t) = x(t)$. How can we calculate $S(t)$ numerically? Do I have to use a delayed ODE solver? Thanks in advance for your help.