# Unable to resolve this linearized equation from a macro growth theory paper

$dk_{it}/dt = (W(\tilde K,\tilde l)/\tilde K)[(1+1/n)(\tilde k_i - \theta_i)(l_t - \tilde l)] + B(k_{it} - \tilde k_i)]$

This is a differential equation of individual capital stock $k_{it}$ linearized arond the steady states of $k_{it}$ & $l_t$. How do I go about finding the stable solution solution to this equation? Considering that there are fewer equations than variables, how is it even possible? (Or am I getting my differential math completely wrong!) The papers states that the stable path is:

$k_{it} = \tilde k_i + (1/(u- B))(W(\tilde K,\tilde l)/\tilde K)[(1+1/n)(\tilde k_i - \theta_i)(l_0 - \tilde l)](k_{it} - \tilde \theta_i)e^{ut}$

Where u is the eigenvalue. Can I get help in deriving this? Just a nudge to right kind of solution method would do, thanks!

Edit: The paper in question.

Search for (A.13) (1st equation) & (A.14) (the stable solution).

• This question would be greatly improved by linking to the paper in question. Nov 8 '16 at 14:20
• Edit: paper in question: vcharite.univ-mrs.fr/PP/penalosa/workingpapers/TurnRamsey1.pdf Search for (A.13) (1st equation) & (A.14) (the stable solution). Going through a lot of his papers, and he's really... terse, at times. Nov 8 '16 at 14:50