Apply Eulers method with the step h 12 to nd an approximati

Apply Euler’s method with the step h = 1/2 to nd an approximation of y(1), where y is the solution of the initial value problem y\' = 1 + y, y(0) = 0.

Solution

Let y_i denote the numerical approximation to the true solution y(x_i) where x_i = ih for integer i > 0.

Let y₀ = y(0) = 1 from the given initial condition. Euler’s method uses f(x_n, y_n), which is the derivative dy/dx evaluated at x = x_n and y = y_n.

For this problem, f(x_n, y_n) = 1+y.

We are required to take two stepsof length 1/2 to determine the numerical solution at x = 1, so we seek the value y₂.

Using Euler’s Method with step size h = 1/2 yields:

y₀ = 0

y₁ = y₀ + hf(x₀, y₀) = 0 + h(1+y₀) = 0 +1/2(1+0) = 1/2

y₂ = y₁ + hf(x₁, y₁) = 1/2 + h(1+y₁) = 1/2 +1/2(1+1/2) = 5/4