Consider the differential equation
\begin{equation*}
\frac{dy}{dx} = 2 x,
\end{equation*}
with initial condition \(y(0) = 3\text{.}\)
A. Use Euler's method with two steps to estimate \(y\) when \(x=1\text{:}\)
\(y(1) \approx\)
(Be sure not to round your calculations at each step!)
Now use four steps:
\(y(1) \approx\)
(Be sure not to round your calculations at each step!)
B. What is the solution to this differential equation (with the given initial condition)?
\(y =\)
C. What is the magnitude of the error in the two Euler approximations you found?
Magnitude of error in Euler with 2 steps =
Magnitude of error in Euler with 4 steps =
D. By what factor should the error in these approximations change (that is, the error with two steps should be what number times the error with four)?
factor =
(How close to this is the result you obtained above?)