= Solution
For $x>0$, take the real contour $\gamma=(-\infty,0)$. At $t\to0^-$ the factor $e^{-1/t^2}$ kills every power, while at $t\to-\infty$ the factor $e^{xt}$ makes the integrand and endpoint term vanish. The <laplace integral solution of a singular third-order equation> is therefore
$$
y(x)=C\int_{-\infty}^0 e^{xt-1/t^2}t^{-3}\,dt.
$$
With $u=-1/t$, and absorbing a minus sign into the arbitrary constant, this becomes
$$
\boxed{y(x)=C_1\int_0^\infty u\exp\left(-u^2-\frac xu\right)du}.
$$
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