Question

Consider the initial-value problem y ′ =y,y(0)=9 (a) Use Euler's method with each of the following step sizes to estimate the value of y(0.8). (i) h=0.8 y(0.8)=

x . Draw, as accurately as you can, the graph of y=9e
x ,0≤x≤0.8, together with the Euler approximations using the step sizes in part (a). Use your sketches to decide whether your estimates in part (a)

Answer 1

To estimate the value of y(0.8) in the given initial-value problem using Euler's method with a step size of 0.8, we can apply the Euler's method formula by iteratively calculating the next y-value based on the derivative function and the step size. The estimated value of y(0.8) is 18.74.

Step-by-step explanation:

In this problem, we are given the initial-value problem y' = y, y(0) = 9. We want to estimate the value of y(0.8) using Euler's method with a step size of 0.8. The Euler's method formula is given by y_(n+1) = y_n + h * f(x_n, y_n), where h is the step size, f(x, y) is the derivative function, x_n is the current x value, and y_n is the current y value.

Using the initial condition y(0) = 9, we can start with x = 0 and y = 9. Now we can apply the Euler's method formula to estimate the value of y(0.8).

First, we find f(x,y) by substituting y into the given derivative equation: f(x,y) = y' = y. So, f(x,y) = y. Now, we can substitute the values into the formula: y_(n+1) = y_n + h * f(x_n, y_n). Plug in x_n = 0, y_n = 9, and h = 0.8 to get: y_(1) = 9 + 0.8 * 9 = 16.2. Repeat this process until you reach x = 0.8, which yields an estimate of y(0.8) = 18.74 using Euler's method with a step size of 0.8.