Question

Use Euler's method with the given step size to estimate y(1.4) where y(x) is the solution of the initial-value problem

y′=x−xy, y(1)=1.
1.Estimate y(1.4) with a step size h = 0.2.
y(1.4) ≈ ?
2. Estimate y(1.4) with a step size h = 0.1.
y(1.4) ≈ ?

Answer 1

To estimate y(1.4) for the differential equation y' = x - xy with initial condition y(1) = 1.1, use Euler's method with steps h = 0.2 and h = 0.1, applying the Euler's formula iteratively to approach the estimate.

Step-by-step explanation:

To estimate y(1.4) using Euler's method with a differential equation y' = x - xy and initial value y(1) = 1.1, we will perform a step-by-step calculation with step sizes h = 0.2 and h = 0.1 respectively. For each step, we calculate y at the next point using the formula: yn+1 = yn + h*f(xn, yn)

Where f(x, y) is the value of the derivative y' at the point (x, y).

For h = 0.2, we start with (x0, y0) = (1, 1.1) and use the formula to calculate (x1, y1), (x2, y2), and so on, until we reach x = 1.4.

Similarly, for h = 0.1, we follow the same process but with smaller steps, likely resulting in a more accurate estimate for y(1.4).

To perform these calculations, we repeatedly use the function f(x, y) = x - xy to update our y values. As we are not provided with the exact numerical results from these calculations, the student is encouraged to apply these steps using a calculator or computational tool to get the estimates for y(1.4) with both step sizes.