# Integrating factor in canonical form of second-order linear equations

In the hyperbolic PDE, I have ticked the part I do not understand. How do they get it to $$v_s(r,s)= r-1 + C(s)e^{-r}$$ in the canonical form process? In the textbook, it's said that they're using some kind of integrating factor method but there is no further elaboration and I am lost here. Can someone explain all the steps in details?

The equation $$(v_s)_r + v_s = r$$ is an ordinary differential equation of the variable $$r$$: $$\frac{\partial y}{\partial r} + y = r ,$$ with $$y(r,s)=v_s(r,s)$$ (the variable $$s$$ may be viewed as a parameter). This leads to solutions of the form $$y=y_p+y_h$$, where $$y_h(r,s) = C(s) e^{-r}$$ is the homogeneous solution. The method of variation of parameter suggests to seek a particular solution of the form $$y_p(r,s) = C(r,s) e^{-r}$$. Finally, one obtains the solutions $$v_s(r,s) = r-1+C(s) e^{-r} .$$