= Solution
Assume the regular Sturm–Liouville setting: real coefficients with $p>0$, $w>0$ and sufficient <continuity> and <differentiability> for the integrations below. Apply a multiplier to $J[y]-\lambda(\int_a^bwy^2-1)$. Its <first variation> in the direction $\eta$ is
$$
2[p y'\eta]_a^b-2\int_a^b\big((p y')'+qy+\lambda wy\big)\eta\,dx.
$$
Variations vanishing at both endpoints and the supplied fundamental lemma give
$$
\boxed{-(p y')'-qy=\lambda wy.}
$$
At a free endpoint the value of $\eta$ is arbitrary, so the natural condition is $p y'=0$ there. At a fixed endpoint $\eta=0$, so no such <derivative> condition is imposed; the prescribed endpoint value is imposed instead. This distinction also allows one endpoint of each type.
For <eigenfunctions> $u,v$ of distinct <eigenvalues> $\lambda,\mu$ under the same homogeneous self-adjoint endpoint conditions, multiply their equations by $v,u$ and subtract. Integration gives
$$
(\lambda-\mu)\int_a^b wuv\,dx=[p(uv'-u'v)]_a^b=0.
$$
The boundary expression vanishes for both natural conditions and homogeneous fixed-value conditions. Thus $\boxed{\int_a^b wuv=0}$ for different <eigenvalues>.
In the specified fixed-endpoint case the eigenproblem is $-y''-y=\lambda y$, $y(0)=y(1)=0$. There are no nonzero solutions when $\lambda+1\le0$; the elementary exponential or affine solutions and the two endpoint conditions prove this. For $\lambda+1>0$, the first condition <leaves of a graph>[leaves] a sine and the second requires frequency $n\pi$. The normalized stationary functions and <eigenvalues> are
$$
\boxed{y_n(x)=\pm\sqrt2\sin(n\pi x),\qquad\lambda_n=n^2\pi^2-1,\quad n\ge1.}
$$
Distinct sine modes are <orthogonal> by the <product-to-sum formula>, or by the preceding <Green second identity>. Although they need not satisfy the natural <derivative> conditions, their fixed zero values also annihilate the boundary expression, so <orthogonality> is expected.
The <Wirtinger inequality> $\int_0^1y'^2\ge\pi^2\int_0^1y^2$ gives $J[y]\ge\pi^2-1$. For completeness it follows by writing $y=v\sin\pi x$ for a smooth function supported away from the endpoints and integrating the cross term: $\int(y'^2-\pi^2y^2)=\int\sin^2(\pi x)v'^2\ge0$. Approximation extends it to the Dirichlet Sobolev class. Equality is attained by the first sine mode; the eigenproblem or equality condition gives only its scalar multiples. Hence the two normalized minimizers are $\boxed{y=\pm\sqrt2\sin\pi x}$ and the <minimum> is $\boxed{\pi^2-1=\lambda_1}$.
There is a wording distinction in the source: the two sign choices of the <minimum> are not <orthogonal> to each other, since their <inner product> is $-1$. <Orthogonality> applies to the stationary modes with different <eigenvalues>, not to both signs of the same minimizer.
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