(Solved): 4. Suppose that y=f(x) is a solution of the differential equation y=xy. The solution of such ...

4. Suppose that $$y=f(x)$$ is a solution of the differential equation $$y^{??}=xy.$$ The solution of such a differential equation has the following properties: 1. $$f$$ is monotone decreasing to 0 on $$[0,?)$$ 2. $$f^{?}$$ is monotone increasing to 0 on $$[0,?)$$ $${}^1$$ In the next exercise, we will establish sharp estimates for $$?(x?a)(x?(a+h))?$$ and $$?(x?a)(x?$$ $$(a+h))(x?(a+2h))?$$. $${}^2$$ We leave this as a challenge question. For those of you who are interested, read page 157 of Numerical Mathematics and Computing, Sixth edition by Kincaid and Cheney. Assume $$f$$ is known at three distinct points \begin{tabular}{l||c|c|c|} $$x$$ & $$a$$ & $$a+h$$ & $$a+2h$$ \\ \hline$$y$$ & $$y_1$$ & $$y_2$$ & $$y_3$$ \end{tabular} where $$a,h>0$$. (a) Find an upper bound on the error produced by the linear interpolating polynomial using the data at $$a$$ and $$a+h$$ over the interval $$[a,a+h]$$. (b) Find an upper bound on the error produced by the quadratic interpolating polynomial using the data at all three points over the interval $$[a,a+2h]$$.