Topic 7.5 Notes – Approximating Solutions Using Euler’s Method

Euler’s Method

Suppose you’re given

$$\frac{dy}{dx} = f(x,y), \quad y(x_0)=y_0.$$

That initial condition gives you a starting point on the solution curve. The differential equation tells you the slope at any point $(x,y)$.

Euler’s idea is simple:

• Start at $(x_0, y_0)$.
• Use the slope there to move a small step forward in $x$.
• Estimate the new $y$-value using the tangent line.
• Repeat.

You’re replacing a curved solution with lots of tiny straight line segments.

The update formula

Everything comes from this one rule:

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

Where:

• $h$ is the step size
• $(x_n, y_n)$ is your current point
• $f(x_n, y_n)$ is the slope at that point
• $x_{n+1} = x_n + h$

You’re adding “slope × run” to the current $y$.

Running the Algorithm

When you see an Euler question on a quiz or FRQ, it usually becomes a table.

Given:

• A differential equation
• An initial condition
• A step size $h$
• A target $x$-value

You repeatedly:

1. Compute the slope $f(x_n, y_n)$.
2. Multiply by $h$.
3. Add to the current $y$.
4. Increase $x$ by $h$.

Quick example

Suppose

$$\frac{dy}{dx} = x - y, \quad y(0)=1, \quad h=0.5.$$

Step 1: Start

• $x_0=0$, $y_0=1$
• Slope: $f(0,1)=0-1=-1$

$$y_1 = 1 + 0.5(-1)=0.5$$

$$x_1=0.5$$

Step 2

• Slope: $f(0.5,0.5)=0.5-0.5=0$

$$y_2 = 0.5 + 0.5(0)=0.5$$

$$x_2=1$$

So $y(1)\approx 0.5$.

Notice how every step uses the new point. That’s a very common place to lose points.

Why It Works

At each step, you’re pretending the function behaves like its tangent line over a small interval.

Here’s the geometric picture:

Euler’s method as repeated tangent-line approximations

Each short line segment shows you following the tangent line from one point to the next by a horizontal distance $h$. At the new point, you recompute the slope and repeat.

This connects directly to:

• Derivatives as slopes (Unit 2)
• Linearization $L(x)=f(a)+f'(a)(x-a)$

Euler’s method is repeated linearization.

Step Size and Accuracy

The step size $h$ controls everything.

• Smaller $h$ → more steps → better approximation
• Larger $h$ → fewer steps → more accumulated error

Error builds at every step because you’re never on the true curve, only on a tangent line.

Overestimate or underestimate?

If the solution curve is:

• Increasing and concave up → Euler typically underestimates
• Increasing and concave down → Euler typically overestimates

That’s because the tangent line sits below or above the curve depending on concavity.

On tests, you may be asked to justify this using concavity or a second derivative argument. Be explicit about why the tangent line lies above or below.

Things That Trip People Up

• Plugging the wrong values into $f(x,y)$. Always use the current $(x_n, y_n)$.
• Rounding too early. Keep several decimals until the final answer.
• Taking the wrong number of steps. Number of steps $=\frac{\text{target }x - x_0}{h}$.
• Forgetting to stop exactly at the requested $x$.

On FRQs, organization matters. A clean table with labeled columns makes scoring easy for the reader.