9.3. Example Problem

Problem: Use Euler’s method to solve 𝑑𝑦/𝑑𝑥 = 𝑥 + 𝑦, 𝑦(0) = 1 for 𝑥 = 0.1, 0.2, 0.3 using step size ℎ = 0.1

In this example, we are tasked with solving a first-order ordinary differential equation (ODE) using Euler’s method. The ODE we have is written as 𝑑𝑦/𝑑𝑥 = 𝑥 + 𝑦, which describes how the variable 𝑦 changes with respect to the variable 𝑥. We know the initial condition, 𝑦(0) = 1, which means that when 𝑥 is 0, the value of 𝑦 is 1. We want to estimate the values of 𝑦 at three points: 0.1, 0.2, and 0.3, using a small step size ℎ of 0.1.

Think of this setup as tracking the height of a plant that grows based on both its current height and the amount of sunlight it received (this is analogous to the term 𝑥 + 𝑦). You start measuring its height at time 0 (when it's 1 meter tall) and want to predict its height after 1 minute, 2 minutes, and 3 minutes while considering how tall it grows over each minute.

Step 1: First iteration (x = 0.1) 𝑦 = 𝑦 + ℎ ⋅ 𝑓(𝑥 ,𝑦 ) = 1 + 0.1(0 + 1) = 1 + 0.1 = 1.1

In the first iteration, we start at the initial point where 𝑥 = 0 and 𝑦 = 1. We apply Euler’s method to estimate the value of 𝑦 at the next point 𝑥 = 0.1. We calculate the slope using the function 𝑓(𝑥, 𝑦) = 𝑥 + 𝑦. For 𝑥 = 0 and 𝑦 = 1, this gives us 𝑓(0, 1) = 0 + 1 = 1. We then plug this into the Euler formula: 𝑦 = 1 + 0.1 * 1, which results in 𝑦 = 1.1.

Imagine you took a small step outside your house, and you measured the change in height of the plant. If the plant is currently 1 meter tall and grows 10 centimeters because of the sunlight at that moment, you can predict its new height (1.1 meters) after that minute.

Step 2: Second iteration (x = 0.2) 𝑦 = 𝑦 + ℎ⋅𝑓(𝑥 ,𝑦 ) = 1.1 + 0.1(0.1 + 1.1) = 1.1 + 0.1(1.2) = 1.1 + 0.12 = 1.22

In the second iteration, we now start with our newly calculated value of 𝑦 = 1.1 at 𝑥 = 0.1. We calculate the slope again using 𝑓(𝑥, 𝑦) with our current values: 𝑓(0.1, 1.1) = 0.1 + 1.1 = 1.2. We apply this to the Euler formula: 𝑦 = 1.1 + 0.1 * 1.2, which results in 𝑦 ≈ 1.22.

Continuing to observe your plant, you check its height again after the second minute. It's now 1.1 meters, and with the sunlight still working its magic, it grows an additional 12 centimeters, resulting in a new height of approximately 1.22 meters.

Step 3: Third iteration (x = 0.3) 𝑦 = 𝑦 + ℎ ⋅ 𝑓(𝑥 ,𝑦 ) = 1.22 + 0.1(0.2 + 1.22) = 1.22 + 0.1(1.42) = 1.22 + 0.142 = 1.362

For the third iteration, we now take our latest found value of 𝑦 = 1.22 when 𝑥 = 0.2. Again, we compute the slope: 𝑓(0.2, 1.22) = 0.2 + 1.22 = 1.42. Using the Euler formula, we calculate 𝑦 = 1.22 + 0.1 * 1.42, leading to 𝑦 ≈ 1.362.

You take another peek at your growing plant after three minutes. With the plant now at around 1.22 meters, it absorbs even more sunlight and stretches to approximately 1.362 meters tall. This growth demonstrates how changes compound over time.

Final Approximate Values: • 𝑦(0.1) ≈ 1.1 • 𝑦(0.2) ≈ 1.22 • 𝑦(0.3) ≈ 1.362

After performing the three iterations using Euler's method, we arrive at our approximate values for 𝑦 at the specified points. Our final results show that 𝑦 is approximately 1.1 when 𝑥 = 0.1, about 1.22 at 𝑥 = 0.2, and approximately 1.362 when 𝑥 = 0.3. These values give us a numerical approximation of how 𝑦 changes as we increment 𝑥.

In the end, just like measuring the height of your growing plant at different intervals, you've collected approximations of its height at each moment (0.1 meters, 0.2 meters, and 0.3 meters), highlighting how it has steadily grown over time.

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