I'm having trouble with part of a question in a problem set. The question reads as follows:

Question

Continuous price discounting. To encourage buyers to place 100-unit orders, your firm's sales department applies a continuous discount that makes the unit price a function p(x) of the number of units x ordered. The discount decreases the price at the rate of $0.01 per unit ordered. The price per unit for a 100-unit order is p(100)=20.09
a. Find p(x) by solving the following initial value problem:
Differential equation: dp/dx= -1/100p
Initial condition: p(100)= 20.09

My first thought was to take the integral of the differential equation and solve for C like so:
-1/100integral of p= (-1/100)((P^2)/2)= -(P^2)/200 +c
20.09= -(100^2)/200 + C
20.09+50= C
c= 70/09
Therefore p(x)= -(p^2)/200+70.09
But this equation doesn't take the discount into account. So I thought maybe the way to find an equation was to use the equation for continous decay:
P=Ce^kt
Where c= the initial value
k= constant
t= number of units?
So we would have:
P=Ce^-.01t
20.09=Ce^(-.01)(100)
C= 54.61 (rounded)
Therefore P=54.61e^.-01t
But this doesn't incorporate the differential equation at all.

Do either of these equations look correct?

Answer 1

price decreases .01 per unit

dprice/dunit= -.01/unit

dprice=-.01 dunit/unit

price= -.01 ln unit + C

Price(100)=-.01 ln 100+ C
C= 20.09 +.01 ln 100
check that.