Describe the Euler's method for solving first-order ODEs. Provide an example.

Euler's method is a numerical technique used to approximate the solution of a first-order ordinary differential equation (ODE). It is based on the idea of approximating the derivative of a function using finite differences.

The general form of a first-order ODE is given by:

dy/dx = f(x, y)

where y is the unknown function and f(x, y) is a given function. Euler's method involves discretizing the domain of the ODE into a set of equally spaced points, and then approximating the derivative at each point using a forward difference.

The Euler's method algorithm can be summarized as follows:

1. Choose a step size, h, which determines the spacing between the points in the domain.
2. Start with an initial condition, y0, which represents the value of the unknown function at the starting point x0.
3. Iterate over the domain, starting from x0 and incrementing by h at each step.
4. At each step, approximate the derivative dy/dx using the forward difference formula:

dy/dx ≈ (y(i+1) - y(i))/h

where y(i) represents the value of the unknown function at the current step, and y(i+1) represents the value at the next step.
5. Update the value of the unknown function at each step using the formula:

y(i+1) = y(i) + f(x(i), y(i)) * h

where f(x(i), y(i)) represents the given function evaluated at the current step.
6. Repeat steps 4 and 5 until the desired number of steps or the desired endpoint is reached.

Here is an example to illustrate Euler's method:

Consider the first-order ODE: dy/dx = x^2 - y

We want to approximate the solution of this ODE using Euler's method with a step size of h = 0.1, starting from x0 = 0 and y0 = 1.

Using the algorithm, we can calculate the approximate values of y at each step:

Step 1: x = 0, y = 1
dy/dx = (0^2) - 1 = -1
y(0.1) = 1 + (-1) * 0.1 = 0.9

Step 2: x = 0.1, y = 0.9
dy/dx = (0.1^2) - 0.9 = -0.89
y(0.2) = 0.9 + (-0.89) * 0.1 = 0.811

Step 3: x = 0.2, y = 0.811
dy/dx = (0.2^2) - 0.811 = -0.764
y(0.3) = 0.811 + (-0.764) * 0.1 = 0.734

Continuing this process, we can calculate the approximate values of y at each subsequent step.

It is important to note that Euler's method provides an approximation of the solution and the accuracy of the approximation depends on the step size chosen. Smaller step sizes generally result in more accurate approximations, but at the cost of increased computational effort.