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  • Task info Open for submissions from Thursday, 28 July 2016, 10:40 PM (1116 days ago)
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Example 1, p. 419

We consider the differential equation \fs2 \epsilon \, y'' +(1+\epsilon)\,y'+y=0 with the boundary conditions \fs2 y(0)=0 and \fs2 y(1)=1. Find the exact solution.

For \fs2 \epsilon\to 0, show that the outer solution satisfies \fs2 y'_{out}+y_{out}=0 with \fs2 y_{out}(0)=0 and the inner solution is such that \fs2 Y''_{in} + Y_{in} =0 with \fs2 Y_{in}(0)=0 and \fs2 Y_{in}(+\infty)=e with \fs2 y_{in}(x) = Y_{in}(X)  and \fs2 x=\epsilon\, X. Draw the matching  between the inner and outer solutions. Write a uniform approximation  under the form \fs2 y_{unif} = y_{in} + y_{out} + y_{match}.