Euler's Method and Direction Fields

Home

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.

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Modeling with Differential Equations

Euler's Method and Direction Fields

Separable Equations

Models of Growth

Linear Equations

Parametrized Curves

Calculus with Parametrized Curves

Polar Coordinates

Areas and Lengths of Polar Curves

Conic sections

Conic Sections in Polar Coordinates

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series

Representing Functions as Power Series

Taylor and Maclaurin Series

Applications of Taylor Polynomials

Functions of 2 and 3 variables

Partial Derivatives

Differentiability and the Chain Rule

Multiple Integrals

Iterated Integrals over Rectangles

Double Integrals over General Regions

Double integrals in polar coordinates

Multiple integrals in physics

Integrals in Probability and Statistics

Change of Variables

Euler's Method

(Or, follow your nose!)

The Method