Question

Expand in Taylor series at both sides to find the error . Sa+"y (3) ds = (34 (2) – y (x – h)) + error

dont use matlab

Answer 1

ANSWER:-

Expand in Taylor series at both side to find the error

By fundamental then On internal calculus

for some func

Y(t) = Ft)

$\therefore \int_{x}^{x+h}Y(s)\,\,ds$

$=\int_{x}^{x+h}F'(s)\,\,ds$

$=F(x+h)-F(x)$

$=F(x)+h\, F'(x)+\frac{h^2}{2!}F"(x)$ for some $C_1$ between
T
and

$= h\, y(x) +\frac{h^{2}}{2!}y'(x) +\frac{h^{3}}{3!}y''c_{1}.......... (1)$as $y(t) =F'(t) \, \forall\, t$

Also, by Tylor series expansion

$=\frac{h}{2}\left ( 3 y(x) -y(x-h) \right )$

$=\frac{h}{2} \left [3 y(x) -\left \{ y(x) -hy'(x) +\frac{h^{2}}{2!}y"(c_{2})\right \} \right ]$

For some C_2 between x-h and x

$=\frac{h}{2} \left [2y(x) +hy'(x) -\frac{h^{2}}{2!}y"(c_{2}) \right ]$

$=h \, y(x)+\frac{h^{2}}{2!}y'(x) -\frac{h^{3}}{2.2!}y"(c_{2}) .......... (2)$

Therefore from equation (1) and (2)

$=\int_{x}^{x+h}y(s) ds-\frac{h}{2}\left [ 3y(x) -y(x-h) \right ]$

$=h^{3}\left ( \frac{y"(c_{1})}{3!}-\frac{y"(c_{2})}{2.2!} \right )$

$=0(h^{3})$

Therefore The error is of $0(h^{3})$