= Solution
The <Sturm-Liouville operator> satisfies the <Lagrange identity for a Sturm-Liouville operator>
$$
y_0L(y;\lambda_0)-yL(y_0;\lambda_0)
=\frac{d}{dx}\bigl[p(y_0y'-yy_0')\bigr].
$$
Indeed the potential terms cancel, and differentiating the right side gives the remaining two derivative terms. Integrate over $[0,1]$. The common <Dirichlet boundary conditions> make the endpoint term zero, while $L(y_0;\lambda_0)=0$. Therefore
$$
\boxed{\int_0^1y_0f\,dx=0.}
$$
This is the necessary <solvability condition> imposed by the nonzero homogeneous solution; its proof here does not rely on merely quoting the <Fredholm alternative>.
For the nonlinear problem, choose
$$
f=L(y;\lambda_0)=y^3-(\lambda-\lambda_0)wy
=y^3-\epsilon^2\mu wy.
$$
The orthogonality condition then gives
$$
\epsilon^2\mu\int_0^1wy_0y\,dx=\int_0^1y_0y^3\,dx.
$$
With $y=\epsilon y_0+\epsilon^2y_1$, where the correction remains bounded as $\epsilon\to0$, the normalization gives
$$
\int_0^1wy_0y\,dx=\epsilon+O(\epsilon^2),\qquad
\int_0^1y_0y^3\,dx=\epsilon^3\int_0^1y_0^4\,dx+O(\epsilon^4).
$$
Divide by $\epsilon^3$ and use $\mu=O(1)$ to obtain
$$
\boxed{\mu=\int_0^1y_0^4\,dx+O(\epsilon).}
$$
The positive quartic integral determines the leading direction of the nonlinear shift from the linear <eigenvalue>.
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