• Use initial conditions to find particular solutions of differential equations.
  • Use slope fields to approximate solutions of differential equations.
  • Use Euler’s Method to approximate solutions of differential equations.

Assignment

  • Vocabulary and teal boxes
  • p375 1–4, 9, 11, 29, 32, 33, 37, 39, 44, 46, 50, 52, 55–57, 59–64 89–93 odd, 99, 101

Additional Resources

We’ll be working with differential equations these next few sections, which involve (x), (y) and the derivatives of (y). One example is below.

\[2xy' - 3y = 0\]

A function is a solution to a differential equation when it, and its derivatives, can be substituted in without issue. For example, (y=e^{-2x}) is a solution to (y’+2y = 0). Since we know (y), we can get (y’=-2e^{-2x}).

\[\begin{align} y'+2y &= 0 \\ \left(-2e^{-2x}\right) + 2\left(e^{-2x}\right) &= 0 \\ 0 &= 0 \end{align}\]

Furthermore, it turns out that anything in the form (y=Ce^{-2x}) is a solution to the differential equation above. This is called the general solution, and the general solution to a differential equation will have a number of constants equal to its order. Order is determined by the highest-order derivative present in the equation. Our example above has an order of one, and so its general solution ends up having one constant in it.

Particular Solutions

Once a general solution has been verified, it can be used to find particular solutions determined by initial conditions. The differential equation (xy’ - 3y = 0) has a general solution of (y=Cx^3) (note the order and number of constants), but we want the solution that passes through ((3,2)). All that’s needed to is to substitute your values into the general solution and solve for (C).

\[\begin{align} y &= Cx^3 \\ (2) &= C(3)^3 \\ \frac{2}{27} &= C \end{align}\]

So, our particular solution is (y = \frac{2}{27}x^3).

Slope Fields

Since general solutions represents all the particular solutions of a differential equation, we use slope fields to simplify viewing an infinite number of curves.

Slope fields are created by first putting the differential in the form (y’=F(x,y)), then plugging in each point into (F(x,y)) to get the slope at that point. This is incredibly tedious to create by hand, so you’ll only do this once or twice in the homework so you get the hang of how they are generated.

Using the ✨magic of technology✨ we can create these pretty easily. My recommendation is GeoGebra. It’s similar to Desmos, but with a lot more power … and complexity. Let’s use the differential from our particular solution to try it out.

Rewriting it, we get (y’=3y/x). So, in GeoGebra enter SlopeField(3y/x) to generate your slope field. And if you enter our particular solution in another entry, you’ll see it match up with the slopes and also pass through ((3,2)).

Finding General Solutions

There are exercises on this topic, but no examples. Luckily, it’s not super complicated, but I will do it slightly differently than the video provided by the book.

\[\begin{align} \frac{dy}{dx} &= 12x^2 \\ dy &= 12x^2 \, dx && \text{Multiply both sides by } dx \\ \int dy &= \int 12x^2 \, dx && \text{Integrate both sides} \\ y &= 4x^3 + C \end{align}\]

It’s going to appear that you can just integrate both sides, in fact all the problems in the section work that way, but things

And apparently I just stopped writing. These notes are from last year and I haven’t reviewed them yet. Sorry if you were getting your hopes up about a thrilling conclusion to slope fields.