Assume that the rate of change of the unit price of a commodity is propirtional to the difference between the demand and the supply, so that
dp/dt = k(D-S)
where k is a constant of proportionality. Suppose that D = 72-5p, S = 9 + 2P, and p(0) = 2. Find a formula for p(t).
dp/dt = k(72-5p-(9+2p))
= k(63-7p)
dp/(9-p) = 7k dt
-ln(9-p) = 7kt + C
1/(9-p) = c*e^(7kt)
9-p = c e^(-7kt)
p = c e^(-7kt) + 9
p(0) = 2, so
c+9 = 2
c = -7
p(t) = 9 - 7e^(-7kt)
To find a formula for p(t), we will first solve the differential equation dp/dt = k(D - S). We can rewrite it using the given values of D and S:
dp/dt = k((72 - 5p) - (9 + 2p))
Now we can simplify it further:
dp/dt = k(63 - 7p)
Separating variables, we move p terms to one side and t terms to the other:
dp/(63 - 7p) = k dt
Next, we integrate both sides with respect to their respective variables:
∫ dp/(63 - 7p) = ∫ k dt
Now we integrate the left side with respect to p:
-1/7 ∫ dp/p - 1/7 ∫ dp/(9 - p) = ∫ k dt
Simplifying the integrals:
-1/7 ln|p| - 1/7 ln|9 - p| = kt + C
To solve for C, we use the initial condition p(0) = 2. Substituting the values:
-1/7 ln|2| - 1/7 ln|9 - 2| = k(0) + C
-1/7 ln(2) - 1/7 ln(7) = C
Simplifying further:
ln(2) + ln(7^(-1/7)) = C
ln(2) - 1/7 ln(7) = C
Now we substitute this value for C back into our equation:
-1/7 ln|p| - 1/7 ln|9 - p| = kt + ln(2) - 1/7 ln(7)
This can be rearranged to:
-1/7 ln|p| - 1/7 ln|9 - p| + 1/7 ln(7) = kt + ln(2)
Combining the logarithmic terms:
-1/7 ln|p(9 - p)| + 1/7 ln(7) = kt + ln(2)
Multiplying through by -7:
ln|p(9 - p)| - ln(7) = -7kt - 7 ln(2)
Using the laws of logarithms to combine the terms on the left side:
ln|p(9 - p)/7| = -7kt - 7 ln(2)
Finally, we exponentiate both sides to isolate p:
|p(9 - p)/7| = e^(-7kt - 7 ln(2))
Taking the absolute value away, we get:
p(9 - p)/7 = e^(-7kt - 7 ln(2))
Multiplying both sides by 7:
p(9 - p) = 7e^(-7kt - 7 ln(2))
Expanding the equation:
9p - p^2 = 7e^(-7kt - 7 ln(2))
Rearranging the terms:
p^2 - 9p + 7e^(-7kt - 7 ln(2)) = 0
This is the final formula for p(t).
To find a formula for p(t), we need to solve the differential equation dp/dt = k(D - S) with the given initial condition p(0) = 2.
Step 1: Substitute the given expressions for D and S into the differential equation:
dp/dt = k((72 - 5p) - (9 + 2p))
Simplifying the expression inside the brackets:
dp/dt = k(72 - 5p - 9 - 2p)
dp/dt = k(63 - 7p)
Step 2: Separate the variables by moving all terms involving p to one side and all terms involving t to the other side:
dp/(63 - 7p) = k dt
Step 3: Integrate both sides with respect to their respective variables:
∫(1/(63 - 7p)) dp = ∫k dt
To integrate the left-hand side, we can use the substitution u = 63 - 7p, which gives du = -7dp:
∫(1/u) du = ∫k dt
ln|u| = kt + C1
Step 4: Solve for u:
ln|63 - 7p| = kt + C1
Step 5: Remove the absolute value by taking the exponential:
e^(ln|63 - 7p|) = e^(kt + C1)
63 - 7p = Ce^kt, where C = ±e^(C1)
Step 6: Solve for p by isolating it:
7p = 63 - Ce^kt
p = (63 - Ce^kt)/7
Step 7: Use the initial condition p(0) = 2 to find the value of C:
2 = (63 - Ce^k(0))/7
14 = 63 - C
C = 63 - 14
C = 49
Step 8: Substitute the value of C back into the equation for p:
p(t) = (63 - 49e^kt)/7
Therefore, the formula for p(t) is p(t) = (63 - 49e^kt)/7.