Question

There are 297 students on a very small college campus, and three students return to campus with the flu. The rate at which the flu spreads among student is given by my. Let ä(t) = the number of students who have the flu at time t, given in days, and y(t) = the number of students who do not have the flu at time t. (a) Rewrite the given ODE with the y variable eliminated. (b) After how many days will 100% students have the flu? Round your number of days up to the nearest whole number.

**correction: the rate at which the flu spreads is given by dx/dt = 1/2xy

(the equal sign is missing in the photo)

Answer 1

(a).

Total number of students = 297

Number of students with flu at time t = x

Number of students without flu at time t = y

2+y = 297

y=297 -

Now , The rate at which flu spreads is given by -

1 d. at TY

Using the value of y in above equation , we have -

-da_1 dt-57(297 - z)

(z:2 — X267) * = =

Thus , the differential equation with eliminated y is -

d. 1902 2 d=5(297T - 22

(b) :

Given the rate of spread of flu is given by -

1 d. at TY

or ,

d. 1902 2 d=5(297T - 22

4P(2.2 – 2267) * = xp

dt = -2 . 72 – 297d2

(1 - 297)

Using integral sign on both sides , we have -

$\Rightarrow \int dt =-2\int \frac{1}{x(x-297)}dx$

Integrating above equation , we get

T--2] ..(1)

Now , to calculate the integral in right , we assume -

1 = TI (2 – 297)

Changing the integrand into partial fraction , we have -

А В TT - 297 (x - 297)

> A2 - 297) + BI (– 297) = 2(x – 297)

=1= A.1 - 297) + Bir

1= AX – 297 A + BC

=1= (A+B). – 297A

COMPARING THE COEFFICIENTS OF BOTH SIDES , WE HAVE-

A+B=0, and -297A = 1

= A=7 and, B =

Thus , we have -

-1/297 1/297 - 297 2.2 - 297)

1 297 1 (x - 297)

Therefore ,

I | , 1 - - 4

And we know that

(1/2)dt = log.

Therefore ,

I = 50% [log x – log(x – 297)

Hence , from eq.(1) , we have -

_- :7 rkܣܐܟ

=-2 x } [log 2 – log(x – 297)]

00-[log 2 – log(2 – 297)

i.e.,

Time (T) is given by -

T = 562[log o – log(x – 297)]

WHERE , x IS THE NUMBER OF STUDENTS WITH FLU .

Therefore , time required when 100 % students get the flu , i.e., x = 297 , is given by -

T = -[log 297 – log(297 – 297)]

[I – +69's] =

[169-740 =

9.388 297

= 0.032 days