Hi, I'm not sure if this is on the right thread but here goes. It's a perturbation type problem.(adsbygoogle = window.adsbygoogle || []).push({});

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!

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# Boundary Value problem and ODE

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