The differential equation below models the temperature of a 95°C cup of coffee in a 21°C room, where it is known that the coffee cools at a rate of 1°C per minute when its temperature is 71°C. How to solve the differential equation?

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Answer 1 · 2017-12-02T13:52:17

See below.

This is a separable differential equation so it can be arranged as

#(dy)/(y-21) = -1/50 dt#

Now, integrating each side

#log_eabs(y-21) = -1/50 t + C_0# or

#y-21 = C_1 e^(-t/50)#

Now, at #t = 0# the temperature is #95^@# or

#95^@-21^@ = C_1 e^0rArr C_1 = 74^@# and finally

#y = 21^@ +74^@e^(-t/50)#