Tutorial Week 3

Worksheet Questions

1. Beer is being poured from a tap into a glass at a rate of $200$ml/min. The beer leaving the tap contains $0.005$g/ml of $\mathrm{CO}_2$. It is known that $\mathrm{CO}_2$ escapes beer at a rate proportional to its concentration, and that at a concentration of $0.1$g/ml, the concentration decreases at a rate of $0.05$g/(ml$\cdot$min).
   1. Write a differential equation with initial conditions for the total amount $Q(t)$ of $\mathrm{CO}_2$ (in grams) in the glass at time $t$.
   2. What would the steady state be if the glass were infinitely tall?
   3. Solve the initial value problem from part (a).
   4. The $\mathrm{CO}_2$ leaving the beer forms a foam. Each gram of $\mathrm{CO}_2$ that escapes produces $40$mm of foam. Additionally, the bubbles burst at a rate of $10\%$ per minute. Write and solve an equation for the height of the foam $F(t)$ at time $t$. (Hint: use your solution from (a) to determine the amount of $\mathrm{CO}_2$ escaping.)
   5. Does the height of the foam $F(t)$ reach a steady state? If so, what is the limiting value?
2. Reduction of Order is a technique for finding a second solution to a homogeneous second order ODE when a first solution has already been found. Consider the equation $t^2y''+ 4ty' + 6y = 0$ where $y_1(t) = t^2$ is our first solution.
   1. Assume the second solution is of the form $y_2(t)=y_1(t)v(t)$ and calculate $4ty_2'(t)$ and $t^2y_2''(t)$.
   2. Plug these in to the ODE and simplify to find a second order equation in terms of $v''(t)$ and $v'(t)$ only.
   3. Rename $v'(t)=w(t)$. Simplify and solve the resulting equation for $w(t)$.
   4. Find $v(t)$ and substitute back to get $y_2(t)$.

Tutorial Week 3 Solutions