Learn application of differential equation using newton’s law of cooling differential equation, compound interest & falling Body Problem.

- Newton's law of cooling says that time rate of change of temperature of a body is proportional to temperature difference between the body and its surrounding medium.

\(\dfrac{dT}{dt} = k \;(T_m - T)\)

where k > 0, T is the temperature of body at any instance 't' , T_{m} is the temperature of surrounding.

- Consider a vertically falling body of mass \(m\) that is being influenced by gravity \('g'\) and air resistance that is proportional to velocity of body.
- Newton's law of motion says that net force acting on body is equal to the time rate of change of momentum of body.

\(F=m\,\dfrac{dv}{dt}\) (\(m\) is constant)

\(\therefore\;mg-kv=m\dfrac{dv}{dt}\)

\(\Rightarrow\;\dfrac{dv}{dt}+\dfrac{kv}{m}=g\;\to\) This is a separable differential equation and can be solved.

- The limiting velocity is defined as \(v_\ell=\dfrac{mg}{k}\) (where resultant force is 0)

A \(28.12\,m/s\)

B \(36.76\,m/s\)

C \(42.13\,m/s\)

D \(52.2\,m/s\)

- Suppose a person places an amount \(N_0\) in a saving account which pays \(P\)% interest per annum compounded continuously, then \(\dfrac{dN}{dt}=\dfrac{P}{100}N\)

where \(N(t)\) is the balance in account at time \('t'\)

\(\Rightarrow\;\dfrac{dN}{N}=\dfrac{P\,dt}{100}\;\to\) Separate the variables

\(\Rightarrow\;ln\,N=\dfrac{P}{100}t+C\)

This equation can be used to find one of the parameter given the value of other.

A \($\,22512.12\;\)

B \($\,23236.38\;\)

C \($\,18192.02\;\)

D \($\,25918.23\;\)

A \(11.55\,\text{%}\)

B \(9.55\,\text{%}\)

C \(10.21\,\text{%}\)

D \(12.25\) %