Let $y_{0}(x)$ be a non-zero solution of the Sturm-Liouville equation

$$L\left(y_{0} ; \lambda_{0}\right) \equiv \frac{d}{d x}\left(p(x) \frac{d y_{0}}{d x}\right)+\left(q(x)+\lambda_{0} w(x)\right) y_{0}=0$$

with boundary conditions $y_{0}(0)=y_{0}(1)=0$. Show that, if $y(x)$ and $f(x)$ are related by

$$L\left(y ; \lambda_{0}\right)=f$$

with $y(x)$ satisfying the same boundary conditions as $y_{0}(x)$, then

$$\int_{0}^{1} y_{0} f d x=0$$

Suppose that $y_{0}$ is normalised so that

$$\int_{0}^{1} w y_{0}^{2} d x=1$$

and consider the problem

$$L(y ; \lambda)=y^{3} ; \quad y(0)=y(1)=0$$

By choosing $f$ appropriately in $(*)$ deduce that, if

$$\lambda-\lambda_{0}=\epsilon^{2} \mu[\mu=O(1), \epsilon \ll 1], \quad \text { and } \quad y(x)=\epsilon y_{0}(x)+\epsilon^{2} y_{1}(x)$$

then

$$\mu=\int_{0}^{1} y_{0}^{4} d x+O(\epsilon)$$