Write and solve the differential equation that models the verbal statement. Using k for the constant of proportionality
The rate of change of N with respect to s is proportional to 626 − s.
dN/ds=
N=
To write and solve the differential equation that models the given verbal statement, we need to express the rate of change of N with respect to s and set it proportional to the expression "626 - s" using the constant of proportionality k.
So, the differential equation can be written as:
dN/ds = k(626 - s)
To solve this differential equation, we will use separation of variables technique:
Step 1: Move the differentials to one side and the terms involving N and s to the other side.
1/(626 - s) dN = k ds
Step 2: Integrate both sides of the equation.
∫ (1/(626 - s)) dN = ∫ k ds
This gives us:
ln|626 - s| + C1 = ks + C2
Where C1 and C2 are integration constants.
Step 3: Simplify the equation.
ln|626 - s| = ks + C
Where C = C2 - C1 is a new integration constant.
Step 4: Eliminate the natural logarithm by taking the exponential of both sides.
e^(ln|626 - s|) = e^(ks + C)
This simplifies to:
|626 - s| = e^C * e^(ks)
Step 5: Remove the absolute value by separating into two cases.
For (626 - s) > 0:
626 - s = A * e^(ks), where A = e^C
For (626 - s) < 0:
s - 626 = A * e^(ks), where A = -e^C
So we have two separate solutions:
Case 1: (626 - s) > 0
N = A * e^(ks) + C3
Case 2: (626 - s) < 0
N = -A * e^(ks) + C4
Where C3 and C4 are integration constants.
These are the solutions to the differential equation that models the given verbal statement, representing the rate of change of N with respect to s, proportional to 626 - s.