Euler's Method and Direction Fields

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dA = r dr (d theta)
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Euler's Method

(Or, follow your nose!)

A differential equation

$\frac{dy}{dx} = f(x,y)$ is like a GPS system: it tells us which way to go if we tell it where we are.

Example: Suppose that

$\displaystyle\frac{dy}{dx} = 0.06 y$ and $y(0) = 1000$ . (This might describe money in a bank account with an initial deposit of $\$$ 1000 and a 6% interest rate.) What will $y(1)$ be?

A reasonable first guess is to say that $\displaystyle\frac{dy}{dx} = 60$ when $x=0$ . Since we're growing at rate 60 , we should expect to grow around 60 between $x=0$ and $x=1$ , so $y(1)$ should be around 1060. That's not far off, but it's not exact. The trouble is that $\displaystyle\frac{dy}{dx}$ isn't constant, as it keeps on changing as $x$ and $y$ change. So although we start growing at a rate of 60, we don't keep growing at that rate.

To get a better estimate we can reduce our step size. Since at first we have $\displaystyle\frac{dy}{dx} = 60$ , we should expect $y(0.5)$ to be around $1000 + (0.5)60 = 1030$ . Then we stop and recompute the derivative. When $y=1030$ , $\displaystyle\frac{dy}{dx} = 61.8$ , so

$y(1) \approx y(0.5) + (0.5) 61.8 \approx 1060.9.$

To get even better, we can take smaller and smaller step sizes.

This is Euler's method!

The Method

In general, the recipe for solving the equation $\displaystyle\frac{dy}{dx} = f(x,y)$ with initial condition $y\left(x_0\right)=y_0$ is:

Pick a small step size. Call it $h$ . We're going to look at times $x_0$ , $x_1=x_0+h$ , $x_2 = x_0 + 2h$ , etc.

Use the differential equation to compute $\displaystyle\frac{dy}{dx}$ at time $x_0$ . Answer: $f\left(x_0,y_0\right)$ .

Use that to estimate $y\left(x_1\right)$ . Call the estimated value $y_1$ .

Use the differential equation to estimate $\displaystyle\frac{dy}{dx}$ at time $x_1$ . Answer: $f\left(x_1,y_1\right)$ .

Use that to estimate $y\left(x_2\right)$ . Call that estimated value $y_2$ .

Rinse, lather, repeat.

The formulas that describe this are:

$x_{n+1} = x_n + h; \qquad y_{n+1} = y_n + h \,f(x_n,y_n).$

The following video goes over this method for the money-in-the-bank example.