Question

Use Euler's method with step size 0.2 to estimate y(1), where y(x) is the solution of the initial-value problem y′=4x+y², y(0)=1.

Answer 1

To estimate y(1) for the given differential equation using Euler's method with a step size of 0.2, the iterative formula y_{n+1} = y_n + h*f(x_n, y_n) is applied, starting from the initial condition y(0) = 1 and proceeding until x reaches 1.

Step-by-step explanation:

To estimate y(1) for the initial-value problem y' = 4x + y², with y(0) = 1, using Euler's method with a step size of 0.2, we follow a step-by-step process:

1. First, identify the formula for Euler's method: y_{n+1} = y_n + h*f(x_n, y_n), where h is the step size, x_n and y_n are the current values of x and y, and f(x, y) is the derivative y'.
2. Substitute the initial values x_0 = 0 and y_0 = 1 into the formula, with h = 0.2 and f(x, y) = 4x + y².
3. Calculate the first approximation y_1 based on the initial values.
4. Repeat this process, incrementing x by the step size at each calculation, until x reaches 1.

The exact values for each step need to be computed using the method outlined above. The provided information does not directly relate to the Euler's method calculations necessary to solve this problem.