# Application of Linear differential equation in solving problems

• MHB
• Help_me1
In summary, after 1 day, 5 students will be aware of the rumour and 10 students will be aware of the rumour after 7 days. It will take 850 students 7 days to be aware of the rumour.
Help_me1
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

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

Since this has been here a while and I just cannot resist:
dr/dt= kr(1000- r)
dr/[r(1000-r)]= kdt

To integrate on the left, separate using "partial fractions"- find constant A and B such that 1/[r(1000- r)]= A/r+ B/(1000- r)
Multiplying on both sides by r(1000- r)
1= A(1000- r)+ Br
Let r= 0: 1= 1000A so A= 1/1000.
Let r= 1000: 1= 1000B so B= 1/1000

1000 dr/r+ 1000 dr/(1000- r)= kdt

The integral of 1000 dr/r is 1000 ln(|r|).
To integrate 1000 dr/(1000- r) let u= 1000- r so that du= -dr.
then dr/(1000- r)= -du/u.
The integral is -1000 ln(|u|)= -1000 ln(|1000- r|)

We have 1000 ln(|r|)-1000 ln(|1000- r|)= kt+ C.
$1000 ln\left(|\frac{r}{1000- r}|\right)= kt+ C$
$ln\left(\left(\frac{r}{1000- r}\right)^{1000}\right)= kt+ C$
(I have dropped the absolute value since this is to an even power.)

Taking the exponential of both sides
$\left(\frac{r}{1000- r}\right)^{1000}= C' e^{kt}$
where $C'= e^C$.

Now, we are given that r(0)= 5 and r(1)= 10.
Setting t= 0, r= 5
$\frac{5}{9995}= \frac{1}{1999}=C'$
Setting t= 1, r= 10
$\frac{10}{9990}= \frac{1}{999}= \frac{e^k}{1999}$
$e^k= \frac{1999}{999}$
so $k= ln\left(\frac{1999}{999}\right)$
and $e^{kt}= (e^k)^t= \left(\frac{1999}{999}\right)^t$

$r(t)= \frac{\left(\frac{1999}{999}\right)^t}{1999}$.

You are putting A and B values in a wrong way ...these values will come in fraction

## 1. What is a linear differential equation?

A linear differential equation is an equation that involves a dependent variable and its derivatives, where the coefficients of the dependent variable and its derivatives are constants. It can be written in the form of y'(x) + p(x)y(x) = q(x), where y'(x) is the derivative of y(x), p(x) and q(x) are functions of x.

## 2. How are linear differential equations used in solving problems?

Linear differential equations are used to model various physical and natural phenomena, such as population growth, radioactive decay, and electrical circuits. By solving these equations, we can predict the behavior of these systems and make informed decisions.

## 3. What are the steps involved in solving a linear differential equation?

The first step is to identify the dependent variable and its derivatives in the equation. Then, we need to separate the variables and integrate both sides. Next, we apply initial or boundary conditions to find the constants of integration. Finally, we can use the solution to make predictions or solve specific problems.

## 4. Can linear differential equations be solved analytically?

Yes, linear differential equations can be solved analytically using various methods, such as separation of variables, integrating factors, and substitution. However, in some cases, it may not be possible to find an exact solution, and numerical methods may be used instead.

## 5. What are some real-life applications of linear differential equations?

Linear differential equations are used in many fields, including physics, engineering, economics, and biology. They are used to model the growth and decay of populations, the flow of electricity in circuits, the cooling of objects, and the spread of diseases. They are also used in the design of control systems and in predicting the behavior of mechanical systems.

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