Section 6.2 Differential Equations: Growth and Decay
Through integration we have been solving differential equations. In other words, we find an equation when given clues about its derivative. In all of the cases we have examined so far we have been able to isolate the derivative. What happens when we cannot isolate the derivative?
Section 6.2 Differential Equations: Growth and Decay The equation below can be solved, but we need to look at a new technique. Look at the equation and think of some way that we can make the problem integrable.
Section 6.2 Differential Equations: Growth and Decay The primary difficulty here is that the x variable is involved with y on the right hand side of the equation. What we need to do here is separate the variables.
Section 6.2 Differential Equations: Growth and Decay The previous equation did not give us any fixed conditions for the equation, so we need to leave it in this general form. How can we find specific solutions? Just like before, we need to know at least one point on the original curve. Let’s try one like that…
Section 6.2 Differential Equations: Growth and Decay A certain curve contains the point (0, 0.5) and its derivative is dy/dx = xy. I’ll be quiet for a couple of minutes while you work on this problem. Feel free to consult your neighbors.
Section 6.2 Differential Equations: Growth and Decay
Finally, we have a class of application problems to consider. These problems involve exponential growth or decay. Populations that grown according to an exponential growth model follow equations that look like
Section 6.2 Differential Equations: Growth and Decay The number of bacteria in a culture is increasing according to the law of exponential growth. There are 125 bacteria in the culture after 2 hours and 350 bacteria after 4 hours. Find the initial population Write an exponential growth model for the bacteria Using your model, predict the population after 8 hours When will the population first exceed 25000?
Section 6.2 Differential Equations: Growth and Decay