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'=3 x+y², y(0)=0 .

y(1)=__

Answer 1

To use Euler's method with a step size of 0.2 for the problem y'=3x+y² with initial condition y(0)=0, we would calculate y values iteratively, but the given details do not correctly demonstrate this process.

Step-by-step explanation:

To estimate y(1) for the initial-value problem y'=3x+y², y(0)=0 using Euler's method with a step size of 0.2, we perform iterative calculations. Starting at x=0 where y=0, we calculate the new values of y at each step by applying the formula yn+1 = yn + step size × f(xn, yn), where f(x, y) = 3x + y² is the given derivative. Continuing this process until x=1 gives us the estimated value of y(1).

However, this detailed explanation does not apply the actual iterative process needed to solve the student's question. The details provided here do not show the appropriate steps for computation and instead offer uncontextual fragments that are irrelevant to Euler's method or the specific problem at hand.