Difference between revisions of "Tutorial Week 3"
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===Worksheet Questions=== | ===Worksheet Questions=== | ||
− | # | + | # 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). |
− | ##Write a differential equation with initial | + | ## Write a differential equation with initial conditions for the total amount $Q(t)$ of $\mathrm{CO}_2$ (in grams) in the glass at time $t$. |
− | ##What would the steady state | + | ## What would the steady state be if the glass were infinitely tall? |
− | ##Solve the initial value problem. | + | ## Solve the initial value problem from part (a). |
− | ## | + | ## 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.) |
− | ## | + | ## Does the height of the foam $F(t)$ reach a steady state? If so, what is the limiting value? |
− | #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 $ | + | #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. |
− | ##Assume the second solution is of the form $y_2(t)=y_1(t)v(t)$ and calculate $ | + | ##Assume the second solution is of the form $y_2(t)=y_1(t)v(t)$ and calculate $4ty_2'(t)$ and $t^2y_2''(t)$. |
##Plug these in to the ODE and simplify to find a second order equation in terms of $v''(t)$ and $v'(t)$ only. | ##Plug these in to the ODE and simplify to find a second order equation in terms of $v''(t)$ and $v'(t)$ only. | ||
− | ##Rename $v'(t)=w(t)$. | + | ##Rename $v'(t)=w(t)$. Simplify and solve the resulting equation for $w(t)$. |
##Find $v(t)$ and substitute back to get $y_2(t)$. | ##Find $v(t)$ and substitute back to get $y_2(t)$. | ||
− | + | [[Tutorial Week 3 Solutions]] | |
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Latest revision as of 23:00, 30 December 2020
Worksheet Questions
- 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).
- Write a differential equation with initial conditions for the total amount $Q(t)$ of $\mathrm{CO}_2$ (in grams) in the glass at time $t$.
- What would the steady state be if the glass were infinitely tall?
- Solve the initial value problem from part (a).
- 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.)
- Does the height of the foam $F(t)$ reach a steady state? If so, what is the limiting value?
- 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.
- Assume the second solution is of the form $y_2(t)=y_1(t)v(t)$ and calculate $4ty_2'(t)$ and $t^2y_2''(t)$.
- Plug these in to the ODE and simplify to find a second order equation in terms of $v''(t)$ and $v'(t)$ only.
- Rename $v'(t)=w(t)$. Simplify and solve the resulting equation for $w(t)$.
- Find $v(t)$ and substitute back to get $y_2(t)$.