Question

Use a numerical solver and Euler's method to obtain a four-decimal approximation of the indicated value. First use h=0.1 and then use h=0.05.

y' = xy² - (x/y), y(1) = 1;y(1.5).

Answer 1

The student's question involves using Euler's method with step sizes of h=0.1 and h=0.05 to numerically approximate the value of y at x=1.5 for the differential equation y' = xy² - (x/y), given the initial condition y(1)=1.

Step-by-step explanation:

The student's question involves using Euler's method to solve a first-order differential equation numerically. To approximate the value of y(1.5) given y' = xy² - (x/y) and y(1) = 1, we can perform Euler's method with step sizes of h=0.1 and h=0.05. Euler's method is a numerical process used to approximate the solutions of differential equations by stepping forward from an initial condition by increments of h, and using the slope of the tangent to the curve at each step to estimate the value of the function.

While the quoted passages provide context for numerical problem solving, they do not directly pertain to the differential equation presented in this question. To apply Euler's method, we increment x in steps of h, updating the y value at each step based on the differential equation's rate of change (y').

Step-by-step, this process involves:

• Calculating the differential equation's slope at the initial condition (x=1, y=1).
• Updating y by adding the product of the slope and step size h to the current y value.
• Repeating these steps until x reaches the desired point (in this case, 1.5).

For a more accurate solution, the process can be repeated with a smaller step size h.