Learn Differential Equation of Exponential Growth and Decay Calculus, practice radioactive decay & exponential growth differential equation.

- In many natural phenomenon quantities grow at a rate which is proportional to their present size.
- If \(y=f(t)\) is population of bacteria at time
**t**, we can expect the rate of growth \(f'(t)\) to be proportional to \(f(t)\), this means that

\(f'(t) \propto f(t)\)

A \(\dfrac{dy}{dt} \propto y\)

B \(\dfrac{dy}{dt} \propto \dfrac{1}{y}\)

C \(\dfrac{dy}{dt} \propto y^2\)

D \(\dfrac{dy}{dt} \propto y^3\)

- In many natural phenomenon quantities grow at a rate which is proportional to their present size.
- If \(y= f(t)\) is population of bacteria at time 't' , we can expect the rate of growth \(f'(t)\) to be proportional to \(f(t)\) , this mean that \(f'(t) = k f(t) \) for some constant k.
- Let \(f(t) = y\) then we have \(\dfrac{dy}{dt} = k\,y \to\) then equation is called the law of natural growth \((k>0)\) .
- It is a differential equation whose solution is very easy to guess.
- The solution to differential equitation \(\dfrac{dy}{dt} = k\,y \) is \(y(t) = y(0) e^{k\,t}\)
- In the context of population growth , if \(P(t)\) is the size of population at time 't' we say \(\dfrac{dP}{dt} = k\,P \) or \(\dfrac{1}{P} \dfrac{dP}{dt} =k\) ,

it is called the relative growth rate or it is growth rate divided by population .

- We say that the relative growth rate of population is constant.

- In many natural phenomenon quantities decay at a rate proportional to their size.
- In general if \(y(t)\) is the value of quantity \(y\) at time \('t'\) and if rate of change of \(y\)with respect to \('t'\) is proportional to its size \(y(t)\) at any time, then

\(\dfrac{dy}{dt} = k\,y\) \((k<0)\)

This is called law of natural decay.

- Radioactive Decay is the process by which some radioactive substances decay by spontaneous emission of radiation.
- If \(m(t)\) is the mass remaining form an initial mass \(m_0\) of substance after time \('t'\), then relative decay rate is \(\dfrac{-1}{m} \dfrac{dm}{dt}\) is constant.

\(\therefore \, \dfrac{dm}{dt} = k\,m\) \((k<0)\)

- The solution to this differential equitation is \(m(t) = m_0\,e^{k\,t}\).
- The time required for half of any quantity to decay is called half life.

A \(8.3426\,mg\)

B \(9.9261\,mg\)

C \(75.1234 \,mg\)

D \(.0567\,mg\)

A \(15.129\,{\text{yrs}}\)

B \(12.246\,{\text{yrs}}\)

C \(5.483\,{\text{yrs}}\)

D \(2.794\,{\text{yrs}}\)