Question

Find the particular solution of the differential equation that satisfies the initial condition. (Enter your solution as an equation.)

Differential Equation:
dT+k(T-60)dt=0

Initial Condition:
T=139 when t=0

Answer 1

T = 79e^(-kt) +60

Explanation:

You want the particular solution to the differential equation ...

dT +k(T -60)dt = 0 . . . . T(0) = 139

Solution

This is a separable differential equation, so we can solve it by separating the variables and integrating each side:

`(dT)/(T-60)=-k\,dt\\\\\displaystyle \int{(dT)/(T-60)}=\int{-k}\,dt\\\\ln((T-60))=-kt+C\\\\T-60=e^(-kt+C)=Ce^(-kt)\qquad\text{take antilogs; values of $C$ are different}`

Applying the initial condition, we have ...

139 -60 = C = 79

Then the particular solution is ...

`\boxed{T=79e^(-kt)+60}`