Last week we introduced differential equations. Today, we looked at lots of examples. We saw that you can often predict the long term behavior of solutions just by analyzing the differential equation (and its slope field) without actually solving it! We looked at the following examples:
\[\displaystyle \frac{dP}{dt} = 1.05 P \left( 1- \frac{P}{10000} \right).\]
At what population level \(P\) would the population stop growing?
Plot the slope field for this equation.
Find \(\lim_{t \rightarrow \infty} P(t)\).
For now, it is too hard to solve logistic growth equations like the one above, but you can still figure out what will happen qualitatively by looking at the slope field.
The current \(I\) in an electrical circuit with a 5 Ohm resistor, a 4 Henry inductor, and a 6 Volt power source will obey the differential equation:
\[4 \frac{dI}{dt} + 5I = 6.\]
Isolate \(dI/dt\) in the equation above.
At what level does the current stop changing?
What will happen to the current \(I\) as \(t \rightarrow \infty\)?
Solve for \(I(t)\). Assume that the initial current is 0 at time zero when the circuit is connected.
The rate of change in the temperature of an object is directly proportional to the difference between the temperature of the object and its surroundings. Translated into an equation, this becomes:
\[ \frac{dT}{dt} = k (T - T_S)\]
where \(T\) is the temperature of the object, \(t\) is the time, \(k\) is a proportionality constant, and \(T_S\) is the temperature of the surroundings.
Graph the slope field for this differential equation. Can we predict the limit of \(T(t)\) as \(t \rightarrow \infty\) without solving?
Solve this differential equation for a hot cup of coffee that is initially \(80^\circ\)C and is left in a room that is \(20^\circ\)C. In order to find \(k\), you’ll need one more piece of info: suppose the coffee is \(40^\circ\)C after 10 minutes.
Today we talked about Euler’s method for approximating solutions to differential equations. Euler’s method can be used to approximate solution of any differential equation of the form \[\frac{dy}{dx} = F(x,y).\] To use Euler’s method, choose a \(\Delta x\). Then repeat the following two steps as many times as needed:
Step 1. Calculate \(\Delta y = F(x,y) \Delta x\).
Step 2. Replace \(x\) by \(x+dx\) and \(y\) by \(y+dy\).
I did several examples using Sage/Python online (see the Examples page). I recommend seeing if you can tweak those examples for the homework this week. Here is a Kahn Academy Video if you would like a different explanation of Euler’s method.
These were the examples we did in class:
\(\dfrac{dy}{dx} = x-y\)
\(\dfrac{dy}{dx} = y\)
\(\dfrac{dP}{dt} = P\left(1- \frac{P}{10000} \right)\)
\(\dfrac{dP}{dt} = r\) where \(r\) is a constant interest rate.
The last example led to a discussion of annual versus monthly versus daily compounding of interest rates. Those different methods are actually just the Euler’s method approximations to continuous compounding with \(\Delta t\) equal to 1 year, 1/12 year, and 1/365 year respectively. This also explains the formula:
\[e^r = \lim_{n \rightarrow \infty} \left(1 + \frac{r}{n} \right)^n\]