= Solution
The condition is that the velocity be the pure irrotational growing-mode velocity, with the homogeneous component in part (c) absent. It is enough to impose $\mathbf u_{\rm in}=-\nabla\psi_{\rm in}$ and exclude additional harmonic or transverse motions through the boundary conditions. The linear equation for $\mathbf u+\nabla\psi$ then keeps that difference zero. A time-independent velocity also forces the same equality in the linear motion equation when its coefficient is nonzero.
At first order the gradient can be evaluated at the initial position: shifting its argument by the first-order displacement changes the velocity only at second order. Integrate $d\mathbf x/db=-\nabla_{\mathbf x_{\rm in}}\psi_{\rm in}$ from $b_{\rm in}$ to $b$, giving
$$
\boxed{\mathbf x=\mathbf x_{\rm in}-(b-b_{\rm in})\nabla_{\mathbf x_{\rm in}}\psi_{\rm in}.}
$$
The trajectory is straight in comoving position as a function of growth time. The physical trajectory $a(t)\mathbf x(t)$ need not be straight or have constant physical <peculiar velocity>. This linear displacement relation is the basis of the <Zeldovich approximation>.
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