Assume a series solution x(t) =summation ^ infinity _ n = 0 a_n t ^n for the equation x" + x = 0. show that the recursion formula for the coefficients is a_m + 2= -a _m/(m + l1(m + 2), m = 0, 1, and that this leads to the general solution x (t) = a_0 (1 - t^2/2! + t^4/4! + +a_1 (t - t^3/3! + t^5/5! + = a_0 cos(t) + a_1 sin(t)

Assume

Substituting these series into the ODE gives

from which we get

for , given and . Now,

and so on, with the general rule

So the series solution is

your answer would be 12

- "runner b is moving at a slower rate."

- "the graphs shown are graphs of distance(km) against time(min).

the slopes of this graphs are are given by:

slope = (change in y)/(change in x)

= (change in distance)/(change in time)

this is the formula for finding the speed. so, the slope of these graphs represent the speed of the runner.

taking each division in the graph to represent 1 on both axis:

runner a ⇒ speed = (2-0)/(1-0)

= 2/1

= 2 km/min

runner b ⇒ speed = (3-0)/(3-0)

= 3/3

= 1 km/min

this shows that   runner b is moving at a slower rate."