The general solution to the differentional eqn dy/dt=ky is y=Ce^kt. This is true becuase the deriv of e^t, and any multiple of it, is itself. when k is positive there is growth, when negative there is decay. The graphs of these solutions are exponential curves with y intercepts that have the x axis as a horizontal asymptote. Population growth, continuously compounded interest, and other instances of growth/decay follow this model.
An equilibrium solution holds true for all y values. The graph is therefore a horizontal line. They are found by setting deriv=0. The equilib func is stable when a small change of initail condions gives a solution that stays close to the equilib as y gets increasingly bigger, but is unstable it veers from the equilib. Newton's law of heating and cooling state that a differently tempertured object will approach room temperature at a rate proportional to thier initial temp, therefore 2 diff temperatured objects will after a given period be very close in temp.
Challenges.
The text states that a solution is a function that is its own derivative. How is this so?
Reflection
Does Newton's law relate to the law of entropy?
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