Euler's Method

A slope field shows the direction a solution curve takes at every point, but not a formula for the curve. Euler's method turns the slope field into numbers: it estimates the value of a solution by walking along short tangent segments, reading a new slope from the differential equation at the start of each one. It works for any differential equation $\frac{dy}{dx} = F(x, y)$, including ones you cannot solve.

One step is a tangent line

Suppose a solution passes through $(x_0, y_0)$. The equation gives its slope there, $F(x_0, y_0)$. For a short run $\Delta x$, the tangent line rises by about $\Delta y = F(x_0, y_0)\,\Delta x$, so

$$y_1 = y_0 + F(x_0, y_0)\,\Delta x \quad \text{at} \quad x_1 = x_0 + \Delta x.$$

This is the local linearization of topic 4.6, with the slope supplied by the differential equation. Euler's method simply repeats it: from $(x_1, y_1)$, read the slope $F(x_1, y_1)$ and step again.

Several steps, in a table

On the exam, show each step: the slope used, the point it was read at, and the new value. Arithmetic does not need simplifying, so $1.5 + 2(0.5)$ earns the point as written. Take exactly the number of steps asked for: a third step overshoots.

Too big or too small?

Each Euler step follows a tangent line, so the shape of the solution decides the error. The figure shows two steps on $\frac{dy}{dx} = y$ from $(0, 1)$. The solution through that point is $y = e^x$ (check: its derivative is itself), which is concave up, so the steps fall below it: Euler gives $2.25$ for a true value of $e \approx 2.718$.

To find the concavity without solving, differentiate the equation implicitly. For $\frac{dy}{dx} = x + y$:

$$\frac{d^2y}{dx^2} = 1 + \frac{dy}{dx} = 1 + x + y,$$

which is positive for $x \ge 0$ and $y > 0$. The solution curves there are concave up, so the estimate 2.5 in worked example 1 is an underestimate. With more than one step, the concavity must hold for every solution curve a step starts on, not only for $f$: each step lies below the curve through its own starting point, and that curve stays below $f$ because solution curves do not cross. Here every step stays where $x \ge 0$ and $y > 0$. Concave down curves lie below their tangent lines, so Euler's method would give an overestimate. Name the concavity and say why it decides the answer; "the solution is increasing" is not a reason.

Step size

Shorter steps update the slope more often and follow the curve more closely. For $\frac{dy}{dx} = y$ from $(0, 1)$, the true value at $x = 1$ is $e \approx 2.718$:

A step can also go left: to estimate $f(0.5)$ from a point at $x = 1$, use $\Delta x = -0.5$.

Euler's method in context

The rate has units of liters per minute and $\Delta t$ has units of minutes, so each $\Delta V$ is in liters, the units of the estimate.