General Solution and Characteristic time

[JoshWalker]

So the question I'm stuck on asks me to find the general solution - I've missed a few lectures so I have no idea what the general solution refers to and I'm having trouble finding concise help online. In addition the second part of the question refers to 'characteristic time' which I also do not know what it refers to.

a) The number of fish in a lake, approximated by a real number x(t), is decreasing according to the differential equation
dx/dt =-x/2
where time is measured in years. Find the general solution to this equation.
b) Find the characteristic time (tau) after which the number of fish decreases by e^-1

If I could get help in the form of explaining the general solution and concise time are in addition to how to apply them in this situation it would be ideal.

 

[HallsofIvy]

You can write dx/dt= -x/2 as dx/x= -(1/2)dt. Integrating both sides ln|x|= -(1/2)t+ C ("C" is the "constant of integration). This is the "general solution" because different values of C gives different solutions and any solution can be written in that form. You could aso take the exponential of both sides get \(\displaystyle |x|= e^{-(1/2)t+ C}= e^{-(1/2)t}e^C= C' e^{-(1/2)t}\) where I have taken \(\displaystyle C'= \pm e^C\) and I can drop the "\(\displaystyle \pm\)" since I can also take C' to be positive or negative. That could also be called the "general solution". Again any value of C' gives a solution and any solution is of that form for some number C'.

Suppose the initial (t= 0) number of fish is A. Then \(\displaystyle A= C'e^{-(1/2)0}= C'\). Since C' is a constant, at any other time, t, we have \(\displaystyle x(t)= Ae^{-(1/2)t}\). For what later time, t, will x have decreased by a factor of \(\displaystyle e^{-1}\)? That is, what is t when \(\displaystyle Ae^{-(1/2)t}= Ae^{-1}\)? We can divide by A getting \(\displaystyle e^{-(1/2)t}= e^{-1}\). Can you solve that for t?