Mathematics

Problem

The temperature

T(t)

(in kelvins) of a chemical reactor satisfies

\frac{dT}{dt}=\frac{t e^{t}}{T},\qquad T>0,

where

t

is measured in hours. a. Find the general solution of the differential equation, expressing your answer in explicit form

T=f(t)

. b. At

t=0

, the temperature is measured to be

300\,\text{K}

. Determine the explicit particular solution that satisfies this initial condition. c. Using your solution from part (b), compute

\displaystyle\frac{d^{2}T}{dt^{2}}

t=0

and interpret its sign in terms of whether the temperature is initially accelerating upward or downward.