Question

problem 1, 2-1, 2-2, 3, 4

and f is nonnegative

A strange way of differential equation solving without know- ing the Fundamental Theorem of Calculus. ! 忑(x) = f(x), 0 < x < 1, Consider a differential equation where f : [0, 1] → R be in C(0,1)) We prove that there is a solution u e C(a,b) of this differential equation without using the fundamental theorem of calculus but using that any continuous function is a limit of piecewise linear functions of finite pieces. We will use Theorems 1 and 2 in the Appendix. Step 1.1 (Divide into n-intervals). Let zo = 0 < x1 < x2く… < xn-1, where xi Let Vi = f(m) for i = 0,.., n. Also h := n n Define a piecewise linear function Ln(x) using the data points, ifェElri,ェi+1), i+i -yi Ti). 7え 1-1土 rri

Answer 1

겨K Ar

2- o we T- 140