Math 2400 – Introduction to Differential Equations – Homework 1
Due date: Friday, January 26 at 11:59 PM on Gradescope
1. Find a function y(t) that solves the initial-value problem
ty ′ + (t + 1)y = 6t,
y(1) = 2,
t ≥ 1.
2. Find an explicit solution of the differential equation
dy
= y(1 − y).
dt
3. To cook a turkey you are to put it into a 350◦ F oven, and cook it until it reaches 165◦ F. In answering
the following questions, assume Newton’s law of cooling is used.
3.1 Suppose the turkey starts out at room temperature, which is 70◦ F. What IVP does the temperature satisfy?
3.2 Suppose that after two hours in the oven, the temperature of the turkey is 140◦ F. How much
longer before it is done?
3.3 Suppose the turkey is taken from the refrigerator, which is set to 40◦ F, and put directly into the
oven. How much longer does it take to cook than when the turkey starts out at room temperature?
(The value for k is the same as in 3.3).
4. A well-mixed open tank initially contains 100 L of water with a salt concentration of 0.1 kg/L. Salt
water enters the tank at a rate of 5 L/h with a salt concentration of 0.2 kg/L. An open valve allows
water to leave at 4 L/h and at the same time water evaporates from the tank at 1 L/h.
4.1 Determine the amount and concentration of salt at any time (that is, as a function of time t.)
4.2 What is the limiting concentration?
5. In each of the following problems, sketch the graph of f (y) versus y, find the steady states, and classify
each one as asymptotically stable, asymptotically unstable, or semi-stable. Draw the phase line, and
sketch several graphs of solutions in the ty-plane. Here y0 = y(0). You should consider concavity to
help sketch the solutions.
5.1
dy
= ay + by 2 , a > 0, b > 0, −∞ < y0 < ∞
dt
5.2
dy
= y(y − 1)(y − 2), y0 ≥ 0
dt
5.3
dy
= −k(y − 1)2 , k > 0, −∞ < y0 < ∞
dt