(a). Solve the eqaution f(x)=0 to find the critcal points of the differetnial eq

Question

(a). Solve the eqaution f(x)=0 to find the critcal points of the differetnial equation dx/dt=f(x):

(b). Dtermine if the critcal point is stable, semi-stable or unstable.

(c). Solve the differnetila eqaution explicity for x(t) in terms of t.

The eqaution to use is:

dx/dt=3-x

Explanation / Answer

(a) f(x) = 0 => 3-x = 0 => x=3 (b) ? (c) dx/dt = 3-x => dx/(3-x) = dt Now integrate both sides to get: -Log(3-x) = t + C where Log means the natural logarithm and C is an arbitrary constant So Log(3-x) =C-t ... remember C is still arbitrary => exp(C-t) = 3-x => x = 3 - exp(C-t) = 3 - C*exp(-t) ... okay since C is still arbitrary i.e. x = 3 - C*exp(-t) [eqation 1] Since x(0)=3 ... see (a), we can plug this initial condition into equation 1 to get 3 = 3 - C*exp(0) = 3 - C = > C = 0 Hence: x(t) = 3 is the solution.

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