= Solution
In one dimension, $u\in W^{1,2}$ has an absolutely continuous bounded representative. For any smooth zero-endpoint <test function> $\eta$, the functions $u+t\eta$ are admissible and arbitrarily close to $u$ for small $t$. Local minimality forces the <first variation> to vanish. Differentiation under the <integral> is valid because $u,\eta$ are bounded and the derivative terms are integrable, giving the weak <Euler-Lagrange equation>
$$
\boxed{\int_0^1\bigl(2u'\eta'+4xu^3\eta\bigr)\,dx=0.}
$$
In distributional form this says $(2u')'=4xu^3$. The fundamental theorem for <weak derivatives>, or the fundamental lemma applied after subtracting an antiderivative, gives a constant $C$ with
$$
\boxed{u'(x)=C+2\int_0^x t\,u(t)^3\,dt\quad\text{almost everywhere}.}
$$
This is the integrated <Euler-Lagrange equation>. The boundary difference $u(1)-u(0)=2$ determines
$$
C=2-2\int_0^1(1-t)t\,u(t)^3\,dt.
$$
The weak identity extends to all zero-trace $W^{1,2}$ <test functions> by <density>.
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