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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?

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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 $$ v_s(r,s) = r-1+C(s) e^{-r} . $$

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