Suppose that we use Euler's method to approximate the solution to the differential equation

𝑑𝑦/𝑑𝑥=𝑥^4/𝑦 𝑦(0.1)=1

Let 𝑓(𝑥,𝑦)=𝑥^4/𝑦.
We let 𝑥0=0.1 and 𝑦0=1 and pick a step size ℎ=0.2. Euler's method is the the following algorithm. From 𝑥𝑛 and 𝑦𝑛, our approximations to the solution of the differential equation at the nth stage, we find the next stage by computing

at (0.1 , 1 )

slope = dy/dx = (0.1)^4 /1 = (10^-1)^4 = 10^-4
at x = x + h = 0.1 + 0.2 = 0.3
Using Euler
our guess at y = 1 + h (dy/dx) = 1 + 0.2 * 10^-4 = 1 +0.00002 = 1.00002

the value of 𝑦 at the next stage using the formula:

𝑦𝑛₊₁ = 𝑦𝑛 + ℎ * 𝑓(𝑥𝑛, 𝑦𝑛)

In this case, 𝑓(𝑥𝑛, 𝑦𝑛) = 𝑥𝑛^4 / 𝑦𝑛.

So, substituting the given values 𝑥0=0.1, 𝑦0=1, and ℎ=0.2 into the Euler's method formula, we have:

𝑦₁ = 𝑦₀ + ℎ * 𝑓(𝑥₀, 𝑦₀)
= 1 + 0.2 * (0.1^4 / 1)
= 1 + 0.2 * (0.00001)
= 1 + 0.000002
≈ 1.000002

Therefore, the approximation for 𝑦 at the first stage is approximately 1.000002.

To find the next stage using Euler's method, follow these steps:

Step 1: Start with the initial conditions 𝑥0 = 0.1 and 𝑦0 = 1.

Step 2: Calculate the slope or derivative 𝑓(𝑥, 𝑦) at the current point (𝑥𝑛, 𝑦𝑛). In this case, 𝑓(𝑥, 𝑦) = 𝑥^4/𝑦.

Step 3: Use the calculated slope to estimate the value of 𝑦 at the next point. The slope can be approximated at the current point as 𝑓(𝑥𝑛, 𝑦𝑛) = (𝑥𝑛)^4/𝑦𝑛. The estimated next value of 𝑦 is given by 𝑦𝑛₊₁ = 𝑦𝑛 + ℎ * 𝑓(𝑥𝑛, 𝑦𝑛), where ℎ is the step size (given as 0.2 in this case).

Step 4: Use the step size ℎ to calculate the next value of 𝑥. The next value of 𝑥 is given by 𝑥𝑛₊₁ = 𝑥𝑛 + ℎ.

Step 5: Repeat step 2 to step 4 with the updated values of 𝑥 and 𝑦 until you reach the desired approximation or the desired number of iterations.

Following this algorithm, you can approximate the solution to the given differential equation using Euler's method.

To use Euler's method to approximate the solution to the given differential equation, we follow these steps:

1. Start with the given initial condition 𝑦(0.1) = 1, 𝑥0 = 0.1, 𝑦0 = 1, and the chosen step size ℎ = 0.2.

2. Compute the slope at the current point (𝑥𝑛, 𝑦𝑛) using the function 𝑓(𝑥, 𝑦) = 𝑥^4/𝑦. In this case, we have 𝑓(𝑥𝑛, 𝑦𝑛) = 𝑥𝑛^4/𝑦𝑛.

3. Use the current slope to approximate the value of the function at the next point (𝑥𝑛+1, 𝑦𝑛+1). The approximation formula for Euler's method is:
𝑦𝑛+1 = 𝑦𝑛 + ℎ * 𝑓(𝑥𝑛, 𝑦𝑛)

4. Repeat steps 2 and 3 until you reach the desired number of steps or the desired endpoint.

Using the above algorithm, we can approximate the solution to the differential equation by iteratively applying Euler's method.

For example:
- Starting with 𝑥0 = 0.1 and 𝑦0 = 1.
- Compute the slope at (0.1, 1) using 𝑓(𝑥, 𝑦) = 𝑥^4/𝑦: 𝑓(0.1, 1) = 0.1^4/1 = 0.00001.
- Approximate the value at the next point (𝑥1, 𝑦1):
𝑦1 = 1 + 0.2 * 0.00001 = 1.000002

Repeat these steps until you reach the desired endpoint or the desired number of steps.

Note: Euler's method provides a first-order approximation, so the accuracy of the solution improves as the step size ℎ decreases.