= Solution
Beyond the outer <classical turning point>, let $\chi(x)=\sqrt{2m[V(x)-E]}$ and $U(x)=\hbar^{-1}\int_{x_2}^x\chi(y)\,dy$. The <bound state> condition removes the growing branch at positive infinity, leaving
$$
\boxed{\psi_{\rm III}(x)\simeq\frac{F e^{-U(x)}}{\sqrt{\chi(x)}},\qquad x>x_2.}
$$
Here $F$ is an arbitrary common amplitude pending the matching and <wavefunction normalization>. For the confining <potential energy> in the native diagram, the decaying branch gives the required exterior <square-integrable function>.
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