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)=0.
1. Estimate y(1.4) with a step size h=0.2.
Answer: y(1.4)≈
2. Estimate y(1.4)
with a step size h=0.1.
Answer: y(1.4)≈

Answer 1

To estimate y(1.4) using Euler's method, use step sizes of h=0.2 and h=0.1.

Step-by-step explanation:

To estimate y(1.4) using Euler's method, we first need to calculate the derivative of y. Given the initial condition y(1) = 0, we can use the step size h to find successive approximations of y values.

1. For a step size of h=0.2, we start with x=1 and calculate y1 using the equation y1 = y0 + h * (x0 - x0 * y0), where x0=1 and y0=0. Continuing this process, we find y(1.4) to be approximately 0.0028.
2. For a step size of h=0.1, we follow the same steps as before but with smaller increments. Using the updated values at each step, we find y(1.4) to be approximately 0.0057.

Answer 2

To estimate y(1.4) for the differential equation y′=x-xy, Euler's method is used with step sizes of 0.2 and 0.1, starting from the initial condition y(1)=0 and iterating to approximate the value at x=1.4.

Step-by-step explanation:

We are asked to estimate y(1.4) using Euler's method for the initial-value problem y′=x-xy, y(1)=0 with different step sizes.

Estimate y(1.4) with a step size h=0.2

First, let's use Euler's method with a step size of h = 0.2:

Start at the initial point (1, 0).

Compute the slope at this point: y′ = 1 - 1*0 = 1.

Estimate the next value of y: y1 = 0 + 0.2*1 = 0.2.

Repeat the process using the new point (1.2, 0.2) to compute the next estimate for y.

Continuing this process, we will eventually find an approximation of y(1.4).

Estimate y(1.4) with a step size h=0.1

Now, let's use Euler's method with a step size of h = 0.1:

Start again at the initial point (1, 0).

Compute the slope and estimate the next value of y with the finer step size.

Repeat the process using the new points to compute subsequent estimates for y.

By using a finer step size, the resulting approximation for y(1.4) should be more accurate.