Differential equation (math)

MAT  275  Online  Activity  4

The equation 𝑦 − 𝑦 − 6𝑦 = 0 has the general solution 𝑦(𝑡) = 𝐴𝑒 + 𝐵𝑒 .

The exponential function has only positive values. Therefore, if 𝐴 and 𝐵 are both positive, then 𝑦(𝑡) is positive for all real numbers 𝑡. Likewise, if 𝐴 and 𝐵 are both negative, then 𝑦(𝑡) is negative for all 𝑡. Things get more interesting when 𝐴 and 𝐵 have opposite signs. For example, the following is a plot of the particular solution

𝑦(𝑡) = −2𝑒 + 𝑒 :

Observe that the solution is negative for all 𝑡 ≥ 0. (We will only consider behavior of solutions on that interval in this activity.)

Another particular solution,

𝑦(𝑡) = −𝑒 + 2𝑒 starts out initially positive, but then switches to negative values, where it remains. The plot illustrates this:

1. Explain, using the (algebraic and limit) laws of the exponential function, why, for any negative 𝐴 and positive 𝐵,

lim →

𝑦(𝑡) = −∞.

2. For each given negative number 𝐴, there is an interval of positive numbers 𝐵 so that the corresponding 𝑦(𝑡) will have a zero for some positive 𝑡. Find that interval for 𝐵, in terms of 𝐴, and the 𝑡 value of the zero in terms of 𝐴 and 𝐵.

3. Can 𝑦(𝑡) have more than one positive zero for a negative 𝐴 and positive 𝐵?