### An apology and modification for workshop #6, Math 151:01-03, spring 2008

I asked for a writeup for problem 4 of workshop #6. I stupidly neglected to write a complete solution before assigning the problem, and therefore did not know then that an additional technique, using an integrating factor, is necessary to find a solution of the differential equation. I very much apologize for this lack of care and regret any difficulty that has occurred. I want to modify the assignment. First here's how the original assignment could be done:

Original assignment

1. Use complete English sentences to describe a mathematical model of the amount of salt, S(t), in pounds, at time t, in minutes after the start of the flow. The resulting model should be a differential equation and an initial condition: an initial value problem.
2. Solve the initial value problem using an appropriate integrating factor.
3. Use the solution of the differential equation to predict the salt concentration at the time that the tank overflows. The student will need to describe how this time is found, and what the concentration is (in appropriate units), rather than the amount of salt.
The second step of the original assignment must be modified (integrating factors are covered in Math 244, taken by engineering and physics and chemistry students, and in Math 252, taken by math majors).

Revised assignment

1. Use complete English sentences to describe a mathematical model of the amount of salt, S(t), in pounds, at time t, in minutes after the start of the flow. The resulting model should be a differential equation and an initial condition: an initial value problem. How students do this is the most important part of the writeup to me.
2. Verify that the function
```       4t2+80t+100
S(t)= -------------
t+10```
is a solution of the initial value problem which was created in the first step. The student must show by explicit computations that this function solves the differential equation and the initial condition. Since the Existence and Uniqueness Theorem guarantees that there will be exactly one solution to the initial value problem, the student's computations show that this function must be the desired solution.
3. Use the solution of the differential equation to predict the salt concentration at the time that the tank overflows. The student will need to describe how this time is found, and what the concentration is (in appropriate units), rather than the amount of salt.
Again, I apologize for my error. Please let me know if you have any questions. I also thank Mr. Priyadashi and various students for bringing this situation to my attention.