Difference between revisions of "Tutorial Week 3"

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##Find $v(t)$ and substitute back to get $y_2(t)$.
 
##Find $v(t)$ and substitute back to get $y_2(t)$.
  
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===Solutions===
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[[Tutorial Week 3 Solutions]]
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##$\dfrac{dQ}{dt}=(0.005 \cdot 200)-0.5Q$  '''(3 points)''', $\ Q(0)=0$ '''(1 point)'''
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##$Q(\infty)=2$g '''(1 point)'''
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##$Q(t)=2 - 2e^{-t/2}$. Integrating factor '''(1 point)''', antiderivatives '''(2 points)''', find Q(t) '''(1 point)''', solve for initial condition '''(1 point)'''
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##$\dfrac{dF}{dt} = 40\cdot0.5 Q(t) - 0.1F$ '''(2 points)''', $F(t) = 100 \left(e^{-\frac{t}{2}}-5 e^{-\frac{t}{10}}+4\right)$ Integrating factor '''(1 point)''', antiderivatives '''(2 points)''', find F(t) '''(1 point)''', solve for initial condition '''(1 point)'''
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##$F(\infty) = 400$mm '''(1 point)'''
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## $4 t y_2' = 4t(2 t v + t^2 v') = 8 t^2 v + 4 t^2 v'$ '''(2 points)''' and $t^2 y_2'' = t^2(2v + 4tv' + t^2 v'')$ '''(2 points)'''
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## $t^4 v''=0$ '''(2 points)'''
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## $w' =0$ '''(1 point)''',  $w(t)= C$ (obvious '''1 point''').
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## $v(t)=Ct + D$ (D=0 is ok) '''(1 point)'''  $ \quad y_2(t) = t^3$ '''(1 point)'''
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Latest revision as of 23:00, 30 December 2020

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

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