9.1 Differential equations (“diff eqs”)

푑푦 A differential equation is an equation that involves at least one derivative (say, 푦′ , or ) of an unknown 푑푥 function 푦. It may also involve the variable x, or the function y, or both.

The order of a diff eq is the order of the highest derivative appearing in the equation. For instance: 푥+5 3푦′ = 2푥2푦 is a first-order differential equation, while = 푦′′ + 7푥 is a second-order diff eq. 푦 We’ll only treat first-order, separable diff eqs in this class. There are many other types of diff eqs that are discussed in more advanced math classes.

To solve a differential equation means to find the function (or family of functions) that satisfy it. This can be difficult, but differential equations arise naturally in many fields, so it’s important to know how to solve them.

Remark: We spent most of this quarter learning how to solve the simplest kind of differential equations, namely those of the form: 푦′ = 푓(푥).

Ex 1 : Find the general solution of the diff eq: 푦′ = 2푥 − 5 :

For harder differential equations, so far the best we can do is guess and check: ln 푥 Ex 2: Is the function 푦 = a solution of the diff eq 푥2푦′ + 푥푦 = 1? 푥

Ex 3: Find a solution of the diff eq 푥2푦′ + 푥푦 = 1 that satisfies the initial condition 푦(1) = 7.

Differential equations arise naturally whenever we have information about the rate of change of a function and want to determine the function itself. For instance, we can model population growth using diff eqs:

Models of Population Growth

1. Assuming ideal conditions (no limits on food or environment, no predators, etc), a population of humans, animals, bacteria, etc grows at a rate proportional to its size. Let t denote the time, P=P(t) the population at time t, then we can model the assumption that “The population grows at a rate proportional to its size” as the diff eq:

Let’s see some consequences implied by this model. Assuming k>0 and P(0)>0 :  What can you say about the sign of the derivative dP/dt?  What does that imply about any solution P?  Graph a few solutions. What does the y-intercept of each graph represent?  Do you know any functions that behave as described? Solve the diff eq.

2. A more realistic model of population growth reflects the fact the environment has limited resources. Normal process is: population grows fast at first, until it gets close to the carrying capacity of the environment, then it levels off. That is, we want our model to satisfy: 푑푃  ≈ 푘푃 for small population P (initially: exponential growth) 푑푡 푑푃  ≈ 0 for P close to carrying capacity K (levels off when population nears K) 푑푡 푑푃  < 0 for P>K (population decreases if over the carrying capacity K) 푑푡

The diff eq for this model (“the logistic diff eq”) was proposed in 1840’s:

푑푃 푃 = 푘푃 1 − 푑푡 퐾

We’ll solve this later (Section 9.4), for now let’s just find out its “equilibrium solutions” (constant solutions: P(t)=C), and sketch some graphs:

9.3: Separable Diff Eqs

푑푦 We say that a diff eq is separable if it’s a first order diff eq that can be written in the form: = 푓 푥 푔(푦), 푑푥 i.e. if the derivative can be written as a product of a function in x only, times a function in y only.

This is a category of differential equations that we can solve:

푑푦 = 푓 푥 푔(푦) 푑푥

If 푔(푦) ≠ 0, then we can separate the variables as follows:

1 푑푦 = 푓 푥 푑푥 푔(푦)

Integrating (the left side with respect to y and the right side with respect to x), we get:

1 푑푦 = 푓 푥 푑푥 푔(푦)

Assuming that we can compute each of the two integrals, we then get an equation in x and y (and constants).

푑푦 Ex 1: Solve the diff eq: = 푦 + 푥푦 푑푥

푑푦 푥 Ex 2: Solve the initial value problem: = , 푦 2 = −3. 푑푥 푦