= Solution
Because the parameter enters through $\lambda wy$,
$$
L(y;\lambda_0)
=L(y;\lambda)-(\lambda-\lambda_0)wy
=y^{m+1}-\varepsilon^m\mu wy.
$$
Use this as $f$ in part (a). The necessary orthogonality condition is
$$
0=\int_0^1y_0\left(y^{m+1}-\varepsilon^m\mu wy\right)dx.
$$
With $y=\varepsilon y_0+\varepsilon^2y_1$ and $\mu=O(1)$,
$$
y^{m+1}=\varepsilon^{m+1}y_0^{m+1}+O(\varepsilon^{m+2}),
\qquad
wy=\varepsilon wy_0+O(\varepsilon^2).
$$
Consequently
$$
0=\varepsilon^{m+1}
\left[
\int_0^1y_0^{m+2}\,dx
-\mu\int_0^1wy_0^2\,dx
\right]
+O(\varepsilon^{m+2}).
$$
The normalization makes the second <integral> one. Divide by $\varepsilon^{m+1}$ to obtain
$$
\boxed{\mu=\int_0^1y_0^{m+2}\,dx+O(\varepsilon)}.
$$
This is the <leading nonlinear eigenvalue shift in a Sturm-Liouville problem>.
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