Problem in finding relationship of ordinary differential equ

[thavamaran]

Hi guys, I am currently working on a set of non-linear ordinary differential equation which is my laser model. Attached is the equations and respective derivation.

I need to find the relationship between N(t) and S(t). At equation (5), if I could find the relationship between this equation to be zero, then I could directly find the N(t) and S(t) relationship. There is relationship between \tau n and \tau p because the theory behind this equations are conservation of energy and definitely there is a relationship between them. Can anyone please advice me on how to solve this final part. Thank you very much!

 

[gtoguy97]

Let's start by introducing some notation. We'll let P(t) denote the total laser power at time t, S(t) denote the energy stored in the laser medium at time t, and N(t) denote the number of photons in the laser cavity at time t.Now, we can derive equations for how P(t), S(t), and N(t) change over time. From conservation of energy, we know thatP(t) = \frac{dS}{dt} + \frac{dN}{dt} (1)From the definition of S(t), we know thatS(t) = \frac{1}{2} \int_0^t P(\tau) d\tau (2)And from the definition of N(t), we know thatN(t) = \int_0^t \frac{1}{\tau_n} P(\tau) e^{-\frac{\tau}{\tau_n}} d\tau (3)where \tau_n is the lifetime of a photon in the cavity.We can use equations (1)-(3) to derive an equation for the relationship between N(t) and S(t). Taking the derivative of equation (2) with respect to t gives us\frac{dS}{dt} = P(t) (4)Substituting equation (4) into equation (1) yields\frac{dN}{dt} = P(t) - \frac{dS}{dt} = P(t) - P(t) = 0 (5)So, equation (5) tells us that the relationship between N(t) and S(t) is that they are both equal to the total laser power P(t).