Asymptotic Analysis and Singular Perturbation Theory

The inner solution near x = −1 is given by
1 + x
y(x, ε) = Y , ε ,
ε
where Y (X, ε) satisfies
Expanding
Y ′′ + (1 − εX)Y ′ + εY = 0,
Y (0, ε) = 1.
Y (X, ε) = Y0(X) + εY1(X) + . . . ,
we find that the leading order inner solution Y0(X) satisfies
The solution is
Y ′′
0 + Y ′
0 = 0,
Y0(0) = 1.
Y0(X) = 1 + A 1 − e −X ,
where A is a constant of integration.
The inner solution near x = 1 is given by
1 − x
y(x, ε) = Z , ε ,
ε
where Z(X, ε) satisfies
Expanding
Z ′′ + (1 − εX)Z ′ + εZ = 0,
Z(0, ε) = 2.
Z(X, ε) = Z0(X) + εZ1(X) + . . . ,
we find that the leading order inner solution Z0(X) satisfies
The solution is
where B is a constant of integration.
Z ′′
0 + Z ′ 0 = 0,
Z0(0) = 2.
Z0(X) = 2 + B 1 − e −X ,
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