Final answer:
To estimate y(0.5) using Euler’s method, we start with the initial condition y(0) = 0 and a step size of 0.1. Using the recursive formula y_(i+1) = y_i + h * f(x_i, y_i), we can iterate the formula 5 times to approximate y(0.5) as 0.0388.
Step-by-step explanation:
Euler’s method is used to approximate the solution of a first-order ordinary differential equation (ODE) with a given initial condition. To use Euler’s method, we start with the initial condition y(0) = 0 and step size h = 0.1. We then use the recursive formula:
yi+1 = yi + h * f(xi,yi)
where f(x,y) is the derivative of the function y(x) given in the initial-value problem. In this case, f(x,y) = 2x * y2. We can iterate this formula to approximate y(0.5) by starting at x = 0 and taking 5 steps.
Step 1: x1 = x0 + h = 0 + 0.1 = 0.1
y1 = y0 + h * f(x0,y0) = 0 + 0.1 * (2 * 0 * 0) = 0
Step 2: x2 = x1 + h = 0.1 + 0.1 = 0.2
y2 = y1 + h * f(x1,y1) = 0 + 0.1 * (2 * 0.1 * 0.1) = 0.002
Step 3: x3 = x2 + h = 0.2 + 0.1 = 0.3
y3 = y2 + h * f(x2,y2) = 0.002 + 0.1 * (2 * 0.2 * 0.2) = 0.0064
Step 4: x4 = x3 + h = 0.3 + 0.1 = 0.4
y4 = y3 + h * f(x3,y3) = 0.0064 + 0.1 * (2 * 0.3 * 0.3) = 0.0188
Step 5: x5 = x4 + h = 0.4 + 0.1 = 0.5
y5 = y4 + h * f(x4,y4) = 0.0188 + 0.1 * (2 * 0.4 * 0.4) = 0.0388
Therefore, the approximation for y(0.5) using Euler’s method with a step size of 0.1 is y(0.5) ≈ 0.0388.