# Boundary Value problem and ODE

 P: 8 Hi, I'm not sure if this is on the right thread but here goes. It's a perturbation type problem. Consider the boundry value problem $$\epsilon y'' + y' + y = 0$$ Show that if $$\epsilon = 0$$ the first order constant coefficient equation has the solution $$y_{outer} (x) = e^{1-x}$$ I have done this fine. Find a suitable rescaling $$X = x/\delta$$ so that the highest derivative is important and balances another term and find the solutions $y_{inner}(X)$ of this equation, containing one free parameter, satisfying the boundary condition at $$x = X = 0$$ So I am at the rescaling part and solved the differential equation (after neglecting the δy part of the full equation) $$\frac{d^2y}{dX^2} + \frac{dy}{dX} = 0$$ yielding $$y = Ae^{-X} + B$$ imposing the boundry condition $$x = X = 0$$ gives $$A = B$$ so is $$y_{inner} = Ae^{-X}$$ ?? I think I covered that the highest derivative is important (Although again I was unsure about the wording) When I continue on I think I either have this part wrong or the matching is wrong as I am not getting the right answer. I am supposed to be doing intermediate scaling. If anyone has any comments about this that would also be much appreciated. Thank you in advance!