`dT + k(T-70)dt = 0 , T(0) = 140` Find the particular solution that satisfies the initial condition

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Given the differential equation : `dT+K(T-70)dt=0, T(0)=140`

We have to find a particular solution that satisfies the initial condition.

 

We can write,

`dT=-K(T-70)dt`

`\frac{dT}{T-70}=-Kdt`

`\int \frac{dT}{T-70}=\int -Kdt`

`ln(T-70)=-Kt+C`  where C is a constant.

Now,

`T-70=e^{-Kt+C}`

         `=e^{-Kt}.e^{C}`

         `=C'e^{-Kt}`    where `e^C=C'` is again a constant.

Hence we have,

`T=70+C'e^{-Kt}`

Applying the initial condition we get,

`140=70+C' ` implies `C'=70`

Therefore we have the solution:

`T=70(1+e^{-Kt})`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial