assume that is the rate of change of the unit price of a commodity is proportional to the diference between the demand and the supply, so that dp/dt=k (D-S) where k is a constant of proportionalily. suppose that D=40-2p, S=5+3p, and P(0)=4.
find a formula for p(t)
To find a formula for p(t), we will first solve the given differential equation:
dp/dt = k(D - S)
Given D = 40 - 2p and S = 5 + 3p, we can substitute these values into the equation:
dp/dt = k((40 - 2p) - (5 + 3p))
Now simplify the equation:
dp/dt = k(35 - 5p)
To solve this separable differential equation, let's rearrange it:
dp/(35 - 5p) = k dt
Now we will integrate both sides of the equation:
∫ dp/(35 - 5p) = ∫ k dt
To calculate the integral on the left-hand side with respect to p, we can use the substitution u = 35 - 5p:
∫ (1/u) du = ∫ k dt
Applying integration, we get:
ln|u| = kt + C1
Now, substituting back u = 35 - 5p:
ln|35 - 5p| = kt + C1
To find the constant of integration, we can use the initial condition P(0) = 4. Plugging in t = 0 and p = 4:
ln|35 - 5(4)| = k(0) + C1
ln|15| = C1
C1 = ln|15|
Finally, the formula for p(t) is:
ln|35 - 5p| = kt + ln|15|
We can solve for p(t) by isolating the variable p:
35 - 5p = e^(kt + ln|15|)
Now, simplify the equation:
35 - 5p = 15e^(kt)
Rearrange it to solve for p:
5p = 35 - 15e^(kt)
p = (35 - 15e^(kt))/5
Simplifying further:
p = 7 - 3e^(kt)
Therefore, the formula for p(t) is p = 7 - 3e^(kt).