Differential Equations Notes

Math 243 - Fall 2025

Week 1 Notes

Mon, Aug 25

We talked about some examples of differential equations.

1. Exponential Growth/Decay. The rate of change in a variable $y$ with respect to time $t$ is proportional to $y$ itself. $$\dfrac{dy}{dt} = k y.$$
   1. Check that $y(t) = Ce^{kt}$ is a solution.
   2. Find the constant $C$ which satisfies the initial value problem with initial condition $y(0) = 1000$.

We talked about dependent and independent variables, the order of a differential equation and how to tell if a function is a solution of a differential equation. We also talked about initial conditions.

2. Spring-Mass Model. The force $F$ of a mass $m$ at the end of a spring can be modeled by Hooke’s Law which says $F = -k x$ where $x$ is the displacement of the spring from its rest position. $$m\dfrac{d^2 x}{dt^2} = -kx.$$

   1. Show that $x = \sin t$ and $x = \cos t$ are two different solutions of the spring equation when $m = k = 1$.
   2. How would the solutions change if $m$ and $k$ were not both equal to 1? How would the oscillation of the spring change if the mass was twice as heavy?
      
   3. How would the spring equation change if there was also a linear drag force $F = -b \frac{dx}{dt}$? What if there is a non-linear drag force $F = -b \left(\frac{dx}{dt}\right)^2$?

The last question led to a discussion of linear versus non-linear differential equations. It’s usually much harder to solve non-linear equations! We will also study systems of differential equations, like the following.

3. Rabbits and Foxes. Suppose there are $R$ rabbits and $F$ foxes in an ecosystem. The rabbit population would grow exponentially, except for the foxes which prey on the rabbits. The fox population would decay exponentially if there wasn’t food to eat, but as long as they can catch enough rabbits, it will grow. The number of rabbits that are eaten by foxes is proportional to the product $RF$.
   $$\begin{align*} \dfrac{dR}{dt} &= a R - b RF \\ \dfrac{dF}{dt} &= -c R + d RF \end{align*}$$

Here is a graph showing these equations as a vector field (with constants $a = 3, b = 4, c = 1, d= 2$).

4. Logistic Growth. $\dfrac{dP}{dt} = kP \left( 1 - \dfrac{P}{N} \right)$ where $k$ is a proportionality constant and $N$ is the carrying capacity.

   1. Under what circumstances does the population $P$ stop changing in this model?
   2. What are the units for the constants $k$ and $N$?
   3. Suppose that we use the logistic growth equation to model a population of rabbits in a region. What if we introduce a predator that always consumes $b$ rabbits per year. How would that change the differential equation above?

Wed, Aug 27

Today we talked about separable equations. We solved the following examples.

1. Solve $\dfrac{dy}{dx} = - x^2 y$.

2. Solve $xy^2 y' = x+1$. (https://youtu.be/1_Q4kndQrtk)

Not every differential equation is separable. For example: $$\frac{dy}{dx} = x-y$$ is not separable.

3. Which of the following differential equations are separable? (https://youtu.be/6vUjGgI8Dso)

   1. $xy' + y = 3$
      
   2. $2x + 2y + 2y' - 1 = 0$
   3. $y' = (x^2+x)(y^2+y)$
   4. $x \dfrac{dy}{dx} + y \dfrac{dy}{dx} = x$

We finished with this example:

4. Newton’s Law of Cooling. The temperature of a small object changes at a rate proportional to the difference between the object’s temperature and its surroundings.

   1. Express Newton’s Law of Cooling as a differential equation.
      
   2. Is that differential equation separable?

5. Mixing Problem. Salty water containing 0.02 kg of salt per liter is flowing into a mixing tank at a rate of 10 L/min. At the same time, water is draining from the tank at 10 L/min.

   1. Write a differential equation to model how the amount of salt in the tank changes with respect to time.
      
   2. Solve the differential equation if the amound of salt in the tank is initially 15 kg. (https://youtu.be/aFfAz9wnoyY)

We didn’t get to this last example in class, but it is a good practice problem.

6. $\dfrac{dy}{dx} = \dfrac{4 \sin x}{3 y^2}$ with initial condition $y(0) = 2$. (https://youtu.be/cc3qtMBdQlE)

Fri, Aug 29

Today we talked about slope fields.

1. Sketch the slope field for $\dfrac{dy}{dx} = x - y$. (https://youtu.be/24pxJ1DuWR8)

Here is a slope field grapher tool that I made a few years ago. You can also use Sage to plot slope fields. Here is an example from the book, with color added.

t, y = var('t, y')
f(t, y) = y^2/2 - t
plot_slope_field(f, (t, -1, 5), (y, -5, 10), headaxislength=3, headlength=3, 
    axes_labels=['$t$','$y(t)$'], color = "blue")

2. Consider the logistic equation with harvesting. $$\dfrac{dP}{dt} = k P \left(1 - \dfrac{P}{N} \right) - h$$ where $h$ is a number of rabbits that are harvested each year.

   1. If $k = 1$, $N = 8$, and $h = 1.5$, then what are the values of $P$ where $\dfrac{dP}{dt} = 0$? Slope field
   2. Graph the function $$y =k x \left(1 - \dfrac{x}{N} \right) - h.$$ Where does the graph cross the x-axis? Is the slope positive or negative at those crossing points?

The logistic equation (with or without harvesting) is autonomous which means that the rate of change $\dfrac{dP}{dt}$ does not depend on time, just on $P$. An equilibrium solution for an autonomous differential equation is a solution where $y'(t) = 0$ for all $t$.

3. In the logistic equation above, what happens to the equilibrium solutions when the rate of harvesting is increased to $h = 2$ and then to $h = 2.5$? What happens to the slope field? What does that mean about the population of rabbits?

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