Calculus
Question 1
(10 Points) Let’s consider the following nonhomogeneous first-order ODE: 3.2y 12t We first investigated how to solve such an ODE with the use of the Linearity Principle, which will be the focus of this question.
a.) Find the solution, yh, to the homogeneous reduction. Be sure to show all work.
b.) Craft an appropriate guess for fir, and find the appropriate coefficient. Be sure to show all work.
c.) Write the general solution to this ODE. d.) Suppose we replaced the -12t in the above ODE with 62t2. What would be an appropriate guess for p? Do NOT solve for it… just tell me what the guess should be and why it is appropri4e.
Question 2
(10 Points) Suppose a bucket is being filled with 2 cups per minute of water, concentrated with 2 grams per cup of sugar. The concentrated water is then being drained at a rate of 3 cups per minute. We assume the bucket is initially filled to the brim with 100 cups of clean water (no sugar at first), and is always well-mixed. The amount of sugar (grams) in the bucket, y, after t minutes can be modeled by: kS
4 100-t
a) Explain how this model was constructed. That is, use the information given to reconstruct this equation. You should be able to explain the origin of every value (e.g. “the number 46 comes from…”)
b.) Using the method of Integrating Factors, solve the ODE.
c.) Since the volume of water entering the bucket is smaller than the volume of water exciting the bucket, the bucket will begin to drain. How much sugar will be in the bucket when the bucket is half full?
