= Solution
Set $X=\epsilon x$. The equation becomes
$$
\epsilon^2y_{XX}+k(X)^2y=0,
\qquad
k(X)=1+X^3.
$$
Since $k$ is positive for every $X\geq0$, there is no <classical turning point>. The leading <WKB approximation for a slowly varying oscillator> is
$$
y\sim\frac1{\sqrt{k(X)}}
\left[
C_+e^{i\epsilon^{-1}\int_0^Xk(S)dS}
+C_-e^{-i\epsilon^{-1}\int_0^Xk(S)dS}
\right].
$$
The condition $y(0)=0$ selects a sine, and $y_x(0)=1$ fixes its coefficient. Since
$$
\frac1\epsilon\int_0^X(1+S^3)\,dS
=x+\frac{\epsilon^3x^4}{4},
$$
the result is
$$
\boxed{
y(x)\sim
\frac1{\sqrt{1+(\epsilon x)^3}}
\sin\left(x+\frac{\epsilon^3x^4}{4}\right)}.
$$
The WKB validity measure
$$
\epsilon\frac{|k'(X)|}{k(X)^2}
=\epsilon\frac{3X^2}{(1+X^3)^2}
$$
is uniformly $O(\epsilon)$ and tends to zero at both ends of $X\geq0$. With no turning point, the approximation is therefore uniformly valid on the entire stated half-line.
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