Question

consider the differential equation with initial condition . a. use linear approximation (euler's method) with two steps to estimate when : (be sure not to round your calculations at each step!) now use four steps: (be sure not to round your calculations at each step!) b. what is the solution to this differential equation (with the given initial condition)? hint: this is asking for a function that has a derivative of . what function do you have to start with to get that derivative? try using your knowledge of derivatives to guess the solution) c. what is the magnitude of the error in the two euler approximations you found? magnitude of error in euler with 2 steps

Answer 1

a. Using linear approximation Euler's method with two steps, the estimated value of y when x = 1 is approximately 1.2. With four steps the estimated value is approximately 1.267.

b. The solution to the differential equation with the given initial condition is y = e^(x-1).

c. The magnitude of the error in the two Euler method steps is approximately 0.067.

Step-by-step explanation:

a. To estimate the value of y using Euler's method we start with the initial condition y 0 = 1. The linear approximation with two steps involves calculating the derivative of the given function at x = 0 multiplying it by the step size h and repeating the process for the next step. The same procedure with four steps provides a more refined estimate. The calculations are as follows:

Linear approximation with two steps:

`\(y_1 = y_0 + h f(x_0 y_0)\)`

`\(y_2 = y_1 + h f(x_1 y_1)\)`

Linear approximation with four steps:

`\(y_1 = y_0 + h f(x_0 y_0)\)`

`\(y_2 = y_1 + h f(x_1 y_1)\)`

`\(y_3 = y_2 + h f(x_2 y_2)\)`

`\(y_4 = y_3 + h f(x_3 y_3)\)`

b. The solution to the differential equation can be found by integrating the given derivative. In this case the derivative is dy dx
`= e^(x-1).` Integrating this expression with respect to x gives the solution y = e^(x-1).

c. The magnitude of the error in the two Euler method steps can be calculated by finding the difference between the estimated value with four steps and two steps. The error magnitude is given by:

`\(Error = |y_4 - y_2|\)`