= Solution
Along the common D-brane worldvolume, the Neumann solution is
$$
X^\mu(\sigma,\tau)=x^\mu+2\alpha'p^\mu\tau
+i\sqrt{2\alpha'}\sum_{n\neq0}\frac{\alpha_n^\mu}{n}
e^{-in\tau}\cos n\sigma,
\qquad \mu=0,\ldots,p.
$$
In each transverse direction, the endpoint conditions $X^I(0,\tau)=x_1^I$ and $X^I(\pi,\tau)=x_2^I$ give
$$
X^I(\sigma,\tau)=x_1^I+\frac{x_2^I-x_1^I}{\pi}\sigma
+i\sqrt{2\alpha'}\sum_{n\neq0}\frac{\alpha_n^I}{n}
e^{-in\tau}\sin n\sigma,
\qquad I=p+1,\ldots,D-1.
$$
The linear term is the classical stretch between the two parallel <D-branes>.
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