= Solution
Take $L=-d^2/dx^2+x$. The <Airy equation> is $Lu=0$. To discuss positivity of this <linear operator>, use its homogeneous variation domain $D(L)=\{v\in H^2(0,1):v(0)=0,\ v'(1)=0\}$. <Integration by parts> gives
$$
\langle v,Lv\rangle_{L^2}=\int_0^1\bigl(|v'|^2+x|v|^2\bigr)dx>0\qquad(v\ne0).
$$
The boundary term is zero at both ends. The same calculation with two different functions proves that the associated form is a <symmetric bilinear form>. Positivity here concerns the homogeneous domain; the condition $u(0)=1$ instead defines an affine set of admissible solutions.
For the <variational problem> define
$$
V=\{v\in H^1(0,1):v(0)=0\},\qquad a(v,w)=\int_0^1(v'w'+xvw)dx,
$$
and minimize
$$
\boxed{J[u]=\frac12\int_0^1(u'^2+xu^2)dx\quad\text{over }u\in1+V.}
$$
The <Sobolev trace> at zero is meaningful in $H^1$. No <derivative> boundary value is imposed on this trial space: the <Neumann boundary condition> will emerge naturally. For $v\in V$, <Cauchy-Schwarz inequality> and integration give
$$
|v(x)|^2\le x\int_0^x|v'(s)|^2ds,\qquad
\|v\|_2^2\le\tfrac12\|v'\|_2^2,
$$
so $a(v,v)\ge\tfrac23\|v\|_{H^1}^2$. Thus $a$ is a bounded, symmetric, <coercive bilinear form> on the <Hilbert space> $V$. The <Lax-Milgram theorem> says that a bounded <coercive bilinear form> and a bounded <linear functional> determine a unique <weak solution>. Apply it to
$$
a(v,w)=-\int_0^1xw\,dx\qquad(w\in V),
$$
and set $u=1+v$. This is precisely the <first variation> condition $a(u,w)=0$ for $J$.
Compactly supported <test functions> first yield $u''=xu$ in the sense of <distributional derivatives>. Since $xu\in L^2$, the solution is in $H^2$. <Integration by parts> then leaves $u'(1)w(1)=0$ for every $w\in V$, hence $u'(1)=0$. The other <boundary condition> is built into $1+V$. Conversely a solution of this <boundary value problem> satisfies the <first variation> condition. Finally,
$$
J[u+w]-J[u]=a(u,w)+\tfrac12a(w,w)=\tfrac12a(w,w)>0\qquad(0\ne w\in V).
$$
This proves existence, uniqueness and the global minimum characterization, including the natural <boundary condition>. It is the <mixed-boundary Airy energy principle>.
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