Question

In: Advanced Math

Ordinary Differential Equations

The number of bacteria in a certain culture grows at a rate that is proportional to the number present. If the number increased from 500 to 2000 in 2 hours, determine.

(a) The number present after 12 hours.

(b) The doubling time.

Solutions

Expert Solution

Solution

(a) The number present after 12 hours.

Let t time after hours

      p the original number of bacteria (t=0)

      x the number of bacteria after time.

\( \implies \frac{dx}{dt}=kx,\quad k\in \mathbb{R} \)

\( \implies\frac{1}{x}dx=kdt \)

\( \implies\int\frac{1}{x}dx=k\int dt \)

      \( ln|x|+c=kt \)

when  \( \begin{cases} t=0\implies x=p,p=500 & \quad \\ t=2\implies x=4p & \quad \end{cases} \)

\( \implies\begin{cases} (1) : ln|p|+c=0\implies c=-ln|p| & \quad \\ (2) : ln|4p|+c=2k & \quad \end{cases} \)

(2) : \( ln|4p|-ln|p|=2k\iff ln\Big(\frac{4p}{p}\Big)=2k\implies k=\frac{ln4}{2} \)

(*) The number present after t=12h

\( \implies lnx-lnp=\frac{ln4}{2}\times 12=6ln4 \)

\( \implies lnx=ln\Big(4^6\times p\Big)\implies x=4^6p=4^6\times 500 \)

(b) The doubling time.

Let T the doubling time , T=2t

\( \implies lnx-lnp=kT=2kt \)

\( \implies lnx=tlnu+lnp \)

\( \implies lnx=ln\Big(4^tp\Big)\implies x=4^tp \)

 


Therefore.

(a). the number of bacteria after time is 500

(b). \( x=4^tp \)

Related Solutions

Ordinary Differential Equations
The rate at which the ice melts is proportional to the amount of ice at the instant. Find the amount of ice left after 2 hours if half the quantity melts in 30 minutes. Solution. Let m be the amount of ice at any time t.
Ordinary Differential Equations
Solve the following problems. \( \frac{d y}{d t} \tan y=\sin (t+y)+\sin (t+y) \)  
Ordinary Differential Equations
Solve the following problems. (a) \( \frac{d y}{d t}=2+\frac{y}{t} \) (b)  \( 3 t+2 y \frac{d y}{d t}=y \)
Ordinary Differential Equations
Solve the following problems.      \( t y \frac{d y}{d t}=\sqrt{t^{2} y^{2}+t^{2}+y^{2}+1} \)
Ordinary Differential Equations
Write a differential equation in wich \( y^2=4(t+1) \) is a solutiion.
Ordinary Differential Equations
Solve the following problems. \( \cos (2 t+y) d y=d t \)
Ordinary Differential Equations
Solve the following problems. \( (2 t-y-1) d y=(3 t+y-4) d t \)
Ordinary Differential Equations
The rate at which a body cools is proportional to the difference between the temperature of the body and that of the surrounding air. If a body in air at 25°C will cool from 100° to 75° in one minute, find its temperature at the end of three minutes.
Ordinary Differential Equations
If the population of a country doubles in 50 years, in how many years will it treble, assuming that the rate of increase is proportional to the number of inhabitants?
Ordinary Differential Equations
Suppose a student carrying a flu virus returns to an isolated college campus of 1000 students. If it is assumed that the rate at which the virus spreads is proportional not only to the number x of infected students but also to the number of students not infected, and assume that no one leaves the campus throughout the duration of the disease, determine the number of infected students after 6 days if it is further observed that after 4 days...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT