consider the initial value problem dy/dx=y^2+1, y(0)=0. estimate y(2)

1) use h=1 to compute your estimate
I got x1=1 x2=2
y=1=1 y2=3
2)use h=1/2 to compute your estimate
I got x1=1/2 x2=1 x3=1.5 x4=2
y1=1/2 y2=9/8 y3=289/128 y4=5.306671143

3) use h=1/8 to compute your answer
there are a lot steps involved here, but I got y 16= 141.22236148

4) what is unusual about your answer? what is going on here and why is it happening?

so my question is why my answers are so different by using differnet value of h? also I don't know how to answer questions 4

To answer why the estimates for y(2) differ based on the choice of h (step size), we can look at the method used to solve the initial value problem: Euler's method.

Euler's method is a numerical technique for approximating solutions to ordinary differential equations. It uses small steps (h) to approximate the values of the solution at different points.

When you choose a larger step size (h=1), the method takes bigger strides between the points where it evaluates the function. This can cause the approximation to be less accurate, as it may overlook significant changes in the function between these points. Hence, the estimate for y(2) obtained with h=1 may not be very precise.

On the other hand, when you choose a smaller step size (h=1/8), the method takes smaller strides, resulting in more points being evaluated and a finer approximation. This generally leads to a more accurate estimate for y(2).

The unusual aspect of your answer with h=1/8 (approximately 141.22236148) is that it deviates significantly from the estimates obtained using larger step sizes. This could be due to the accumulation of error in the approximation method. If the step size is too small, the computation may accumulate numerical errors, leading to a less accurate estimate. This phenomenon is known as "round-off error" and is common in numerical approximations.

To mitigate round-off error, it is often useful to strike a balance between precision and computational efficiency by choosing an appropriate step size. Trial and error can help in finding a step size that yields a good balance between accuracy and computational effort.

For question 4, the unusual aspect mentioned above is what is being referred to. The fact that the estimate obtained with h=1/8 is so different from the estimates obtained with larger step sizes indicates that there may be some numerical issues in the approximation. This is an opportunity to discuss the limitations of numerical methods and the need to choose an appropriate step size for accurate results.