Answer
$$p\left( t \right) = - 10{e^{ - t}} + 110$$
Work Step by Step
$$\eqalign{
& p'\left( t \right) = 10{e^{ - t}} \cr
& p\left( t \right) = \int {p'\left( t \right)} dt \cr
& then \cr
& p\left( t \right) = \int {10{e^{ - t}}} dt \cr
& find{\text{ the general solution}} \cr
& p\left( t \right) = - 10{e^{ - t}} + C \cr
& {\text{using the initial condition }}p\left( 0 \right) = 100 \cr
& 100 = - 10{e^{ - 0}} + C \cr
& 100 = - 10e + C \cr
& C = 110 \cr
& {\text{the solution to the initial value problem is}} \cr
& p\left( t \right) = - 10{e^{ - t}} + 110 \cr} $$
