Question

Consider y′+2xy = x,y(0) = 8. Find y(0.5) and y(1) using Euler's method with h = .01, using the improved Euler method with h = .05, the Runge-Kutta method with h =.1, and by actually solving the differential equation.

Answer 1

To find y(0.5) and y(1) using Euler's method, improved Euler method, and Runge-Kutta method with different step sizes.

Step-by-step explanation:

Euler's method is a numerical method used to approximate the solution of a differential equation. To find y(0.5) and y(1) using Euler's method with h = .01, we will start with the initial condition y(0) = 8.

Using the formula y_{n+1} = y_n + h * f(x_n, y_n), where f(x, y) = -2xy + x, we can iterate to find the values of y(0.5) and y(1).

Similarly, we can use the improved Euler method with h = .05 and the Runge-Kutta method with h = .1 to find the values of y(0.5) and y(1).