s19_2360_material

A friend of mine in college claimed that hot water freezes faster than cold water. His reasoning was that hot water loses heat at a faster rate than cold water, so if you put a container of hot water and a container of cold water in a freezer at the same time, the hot water will cool down faster and will freeze first. What does our heating/cooling model have to say about my friend's theory?

The Snowplow Problem: One night it started snowing at a steady rate. A snowplow started out at midnight. At 1 a.m., it had gone 2 miles. At 2 a.m., it had gone another mile. What time did it start snowing?

The standard form of a second-order linear d.e. is y″ + p(x) y′ + q(x) y = f(x). Recalling the solution procedure for a first-order linear d.e., you might think that there is an integrating factor ρ(x) with the property that, if you multiply both sides of the d.e. by ρ(x), the left side becomes the second derivative of ρ(x)y.
- Show that this is possible only if q(x)=½ p′(x) + ¼ [p(x)]²
- Find a formula for ρ(x) if q(x)=½ p′(x) + ¼ [p(x)]²

For people who have studied linear algebra:
- Show that the set of solutions to the linear homogeneous d.e. y'' + p(x) y' + q(x) y = 0 is a subspace of the set of all twice differentiable functions.
- Let y₁ denote the solution of the IVP y'' + p(x) y' + q(x) y = 0, y(0) = 1, y'(0) = 0 and let y₂ denote the solution of the IVP y'' + p(x) y' + q(x) y = 0, y(0) = 0, y'(0) = 1. (Theorem 2 in section 3.1 of Edwards & Penney's text guarantees that each of these IVP's has a unique solution on any interval on which the functions p and q are continuous.) Show that the set {y₁ , y₂} is a basis for the subspace in part a. Hint: Show that {y₁ , y₂} is linearly independent, and show that {y₁ , y₂} spans the subspace by expressing the solution of the IVP y'' + p(x) y' + q(x) y = 0, y(0) = a, y'(0) = b as a linear combination of y₁ and y₂.

The total energy E of a mass-spring system is given by E = ½mv² + ½kx². Show that E is constant for an undamped, unforced system, and show that E is a decreasing function of time for a damped, unforced system.

Steve Pennell | (978) 934-2710 | stephen_pennell@uml.edu

Last Updated: April 16, 2019

Engineering Differential Equations MATH.2360-201