What is the solution for y ( t ) = e − t τ y ( t ) ?A simple quadratic flow model leads to the following apparently simple equation y ( t ) = e − t τ y ( t ) where the flow, y is a function of time, t and τ is a constant.But is there a closed form solution for y(t) just in terms of t and τ?Or can this only be solved by numerical means ?

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What is the solution for   y  (  t  )  =      e          −              t                  τ          y          (          t          )                    ?A simple quadratic flow model leads to the following apparently simple equation  y  (  t  )  =      e          −              t                  τ          y          (          t          )                    where the flow, y is a function of time, t and τ is a constant.But is there a closed form solution for y(t) just in terms of t and   τ?Or can this only be solved by numerical means ?

mentest91k99 · Accepted Answer

Step 1We begin with the equation                    (1)                    y        (        t        )        =                  e                      −            t                          /                        τ            y            (            t            )                              Let   z  =  −  t      /    τ and let   W  =      z          y      (      t      )      . Then, upon rearranging (1) we find that                    (2)                    z        =        W                  e          W                    Step 2Noting that (2) defines the Lambert W, we have immediately that            y      (      t      )      −                        −          t                      /                    τ                          W                      (            −            t                          /                        τ            )                              Special attention is required to determine the domain for which y(t) is real and single valued.

What is the solution for y(t)=e^{-t/(tau y(t))}?

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