# Application of Linear differential equation in solving problems

#### Help_me

##### New member
A rumour spreads through a university with a population 1000 students at a rate proportional to the product of those who have heard the rumour and those who have not.If 5 student leaders initiated the rumours and 10 students are aware of the rumour after one day:-
i)How many students will be aware of the rumour after 7 days.
ii)How long will it take for 850 students to hear the rumour

#### skeeter

##### Well-known member
MHB Math Helper
let $r$ = number of students who have heard the rumor

$(1000-r)$ = number who have not heard the rumor

$t$ is in days

$\dfrac{dr}{dt} = k \cdot r(1000-r)$, where $k$ is the constant of proportionality

you are given $r(0)=5$ and $r(1) = 10$

see what you can do from here …

Thank you