Please Find the equilibrium points of non Autonomous equatio

Please Find the equilibrium points of non- Autonomous equations?

dx/dt = -x_0(a/bt + 1) - xg - N dy/dt = 150 x^2/3/y dz/dt = 10z (a/bt + 1) + 0.00018 z(e^(-0.000102t))/[0.51 + 0.49(e^(-0.000102t))]^2 where a = 1.6 b = 35.4 X_0 = 100 g = 0.0028 N = 690, 314, 240, 000

Solution

Please note that all these 3 equations are Non-autonomous differential equations. In general, there are no particular equilibrium Points of Non-autonomous differential equations except the trivial ones (0 or infinity)

Further Explanations :
For example, Let\'s assume that we need to find the equlibrium points for following differential equations

dy/dt = f(y, t)

Since t appears explicitly in on the right hand side of the equations , this cannot be a autonomous equation. Therefore, this is a form of autonomous equation.

By definition, an equilibrium point is a value of y such that dy/dt = 0 for all t. If t appears explicitly in f(y,t) then it should be clear that dy/dt = 0 for specific values of t (if at all).

Therefore, only the trivial solutions to non-autonomous differential equations are the equilibrium solutions, but they are usually not of interest.

So since we have only finite values of time for the non-autonomous differential equations rather than a continuous values (let\'s say for t> some t\' we have dy/dy = 0 ) therefore, we cannot have equilibrium solutions for such equations.