Tutorials

Tutorial 1: Review of Algebra and Calculus

Outcomes: At the end of this tutorial you should be able to

- Understand the rules of the natural exponent and logarithmic functions.
- Solve equations containing these functions.
- Differentiate and integrate functions using the associated rules.
- Differentiate and integrate functions using the associated methods

Logarithmic Equations

Solve each of the following logarithmic equations for \(x\):

  1. \(\ln{(5x+4)}=0\)

  2. \(\ln{x}+\ln{0.2}=\ln{e}\)

  3. \(\ln{4x^2}-\ln{16}=\ln{4}-\ln{2x}\)

Revision of Differentiation and Integration Techniques

Differentiate the following functions:

  1. \(x(7x+8)^{2}\)

  2. \(\dfrac{5x^{2}}{(2x^{3}+4)^{4}}\)

Integrate the following functions:

  1. \((2x+1)\ln(x+1)\)

  2. \(x e^{x^{2}}\)

Multiple Choice Questions

  1. If \(f(x)=(x^2-3x)^6(4-x)^5\), then \(f'(x)\) is equal to:

\(\text{(A) } (x^2-3x)^5(4-x)^4(5x^2-21x+24)\)

\(\text{(B) } -(x^2-3x)^5(4-x)^4(7x^2-51x+72)\)

\(\text{(C) } -(x^2-3x)^5(4-x)^4(17x^2-81x+72)\)

\(\text{(D) } 6(x^2-3x)^5(2x-3)(-5)(4-x)^4\)

\(\text{(E) none of these}\)

2) If \(y=e^x \ln{x}\), then \(\dfrac{dy}{dx}\) is equal to:

\(\text{(A) } \dfrac{e^x(1+x\ln{x})}{x}\)

\(\text{(B) } e^x(x+\ln{x})\)

\(\text{(C) } xe^x\)

\(\text{(D) } \dfrac{x}{e^x}\)

\(\text{(E) none of these}\)

3) If \(\dfrac{500}{12+5e^{-0.5x}}\), then \(\dfrac{dy}{dx}\) is equal to:

\(\text{(A) } 500(-1)(12+5e^{-0.5x})^{-2}\)

\(\text{(B) } 500(-1)(12+5e^{-0.5x})^{-2}(12+5e^{-0.5x})\)

\(\text{(C) } 500(-1)(12+5e^{-0.5x})^{-2}(-2.5e^{-0.5x})\)

\(\text{(D) } 500(-1)(5(0.5)e^{-0.5x})^{-2}\)

\(\text{(E) none of these}\)

4) \({\displaystyle \int \dfrac{x^2+4x-\sqrt{x}}{x^2} dx}\), expressed in terms of an arbitrary constant \(c\) is equal to:

\(\text{(A) } x+\ln{4x}-\frac{2}{3}x^{\frac{3}{2}}+c\)

\(\text{(B) } x+\ln{4x}+\dfrac{2}{\sqrt{x}}+c\)

\(\text{(C) } x+4\ln{x}+\dfrac{1}{2\sqrt{x}}+c\)

\(\text{(D) } x+4\ln{x}+\dfrac{2}{\sqrt{x}}+c\)

\(\text{(E) none of these}\)

5) \({\displaystyle \int \left(\dfrac{3x}{2}-\dfrac{9}{4}\right) e^{x(x-3)} dx}\), expressed in terms of an arbitrary constant \(c\) is equal to:

\(\text{(A) } \dfrac{3}{4}e^{x^2-3x}+c\)

\(\text{(B) } e^{x^2-3x}+c\)

\(\text{(C) } \dfrac{9}{4}e^{x^2-3x}+c\)

\(\text{(D) } \dfrac{3}{2}e^{x^2-3x}+c\)

\(\text{(E) } \dfrac{4}{3}e^{x^2-3x}+c\)

6) \({\displaystyle \int (3x+1)^{2} e^{-2x} dx}\), expressed in terms of an arbitrary constant \(c\) is equal to:

\(\text{(A) } -0.5(3x+1) e^{-2x}+1.5(3x+1)^{2} e^{-2x}-2.25 e^{-2x} + c\)

\(\text{(B) } -0.5(3x+1)^{2} e^{-2x}+1.5(3x+1) e^{-2x}+2.25 e^{-2x} + c\)

\(\text{(C) } -0.5(3x+1)^{2} e^{-2x}-1.5(3x+1) e^{-2x}-2.25 e^{-2x} + c\)

\(\text{(D) }-0.5(3x+1)^{2} e^{-2x}+1.5(3x+1) e^{-2x}-2.25 e^{-2x} + c\)

\(\text{(E) none of these}\)

Tutorial 2: Classification, Solutions, and Direction Fields

Outcomes: At the end of this tutorial you should be able to

- Explain what an ordinary differential equation is, and why classifying one is a useful first step.
- Classify a given differential equation by its order, linearity, homogeneity, and coefficients, and justify the classification.
- Verify by substitution whether a given function is a solution of a differential equation.
- Sketch a family of solution curves.
- Sketch and interpret a direction field for a first-order equation, and relate it to the family of solution curves without solving the equation.

Conceptual Questions

  1. Explain the ways in which one can classify a differential equation, and why classification matters for what comes after it.
  2. What does the degree of a differential equation inform us about its linearity?
  3. In your own words: what does a single lineal element at a point \((x,y)\) tell you, and where does that information come from?

Classification of ODEs

Classify the following differential equations by checking first if they are ODEs, and then in terms of their order, linearity, homogeneity, and coefficients. Give reasons for your answers.

  1. \(t \dfrac{d^3 x}{dt^3}-2\left(\dfrac{d x}{dt}\right) ^4+x=0\)
  2. \(y y'+2y=1+x^2\), where \(y=y(x)\)
  3. \(y''+9y=\sin y\), where \(y=y(x)\)
  4. \(\dfrac{d^2 R}{dt^2}=\dfrac{\kappa}{R^2}\), where \(\kappa\) (read: kappa) is a constant
  5. \(x^5\dfrac{d^4 y}{dx^4}-x^3 \dfrac{d^3 y}{dx^3}+6y=0\)
  6. \(\dfrac{d^2 x}{dt^2}=\sqrt{1+\left( \dfrac{d x}{dt}\right) ^2}\)

Solutions of ODEs

  1. In the questions below, verify that the indicated function is an explicit solution of the given differential equation, where \(y=y(x)\).
      1. \(2y'+y=0; \quad y= e^{-x/2}\)
      1. \(y''+y=\tan x; \quad y=-(\cos x)\ln(\sec x+\tan x)\)

  2. In the question below, verify that the indicated family of functions is a solution of the given differential equation.         \(x^3\dfrac{\textrm{d}^3 y}{\textrm{d}x^3}+2x^2 \dfrac{\textrm{d}^2 y}{\textrm{d}x^2}-x\dfrac{\textrm{d} y}{\textrm{d}x}+y=12x^2; \quad y=c_1x^{-1} +c_2x+c_3x \ln x+4x^2\),         where \(c_1\), \(c_2\), \(c_3\) are arbitrary constants.

  3. Draw a rough sketch or use a graphing calculator (like Desmos) to find the family of solution curves for the differential equation \(y'=\cos x; \quad y=\sin x + c\), where \(y=y(x)\).

Direction Fields

  1. Consider the differential equation \(\dfrac{dy}{dx} = y - x\).
      1. Compute the slope \(f(x,y)\) at each of the nine grid points \(x, y \in \{-1, 0, 1\}\), and record your values in a small table.
      1. Using your table, sketch the direction field on this grid by hand.
      1. On your sketch, lightly trace the solution curve that passes through the point \((0, 0)\), following the lineal elements. Do the same through \((0, 1)\). Describe, in a sentence, how the two curves behave differently as \(x\) increases.

  2. Without computing a single slope, match each differential equation below to the description of its direction field. Justify each choice in one sentence.
      1. \(\dfrac{dy}{dx} = 2\)
      1. \(\dfrac{dy}{dx} = x\)
      1. \(\dfrac{dy}{dx} = y\)
      1. \(\dfrac{dy}{dx} = -\dfrac{x}{y}\)

    1. The lineal elements are identical along every horizontal line, flat on the \(x\)-axis, and steepen as you move away from it.
    2. The lineal elements everywhere lie tangent to circles centred at the origin.
    3. Every lineal element in the entire plane has the same slope.
    4. The lineal elements are identical along every vertical line, flat on the \(y\)-axis, and steepen as you move away from it.

  3. Consider the differential equation \(\dfrac{dy}{dx} = 1 - y\).
      1. Sketch the direction field by hand on the region \(x \in [-2, 2]\), \(y \in [-1, 3]\) (a grid spacing of \(1\), or \(0.5\) if you are feeling thorough, is fine).
      1. There is one constant function that solves this differential equation. Read it off your direction field, and verify it by substitution.
      1. Using your field, describe the long-run behaviour (as \(x \to \infty\)) of the solution starting at \((0, 3)\), and of the solution starting at \((0, -1)\). What single feature of the field makes both answers obvious?
      1. Check your sketch and your answers using the interactive direction field widget in the notes (the equation \(y(1-y)\) in the widget’s menu behaves similarly near its constant solutions — explore both).

  4. Connecting the ideas. The differential equation \(y' = \cos x\) from Question 3 of the Solutions of ODEs section has the family of solutions \(y = \sin x + c\).
      1. What special structure does the direction field of \(y' = \cos x\) have, given that the slope does not depend on \(y\)?
      1. Use that structure to explain, in one or two sentences, why every member of the family \(y = \sin x + c\) is just a vertical shift of every other member — and why the “+ \(c\)” from integration was therefore inevitable.

  5. True or false, with a reason: if the direction field of a first-order differential equation \(\frac{dy}{dx} = f(x,y)\) (with \(f\) well-behaved) shows a lineal element of slope zero at the point \((a, b)\), then the constant function \(y = b\) must be a solution of the differential equation.

Tutorial 3: Direct Integration, Separation of Variables, and Initial Value Problems

Outcomes: At the end of this tutorial you should be able to

- Classify a first-order or higher-order ODE and justify which solution method (direct integration or separation of variables) applies --- including when manipulation is needed first.
- Solve a given ODE using the Method of Direct Integration or the Method of Separation of Variables, identifying any constant solutions that division temporarily discards.
- Solve an initial value problem: obtain the general solution first, then use the initial condition(s) to determine a particular solution, selecting the correct branch where relevant.

Solutions of Differential Equations

Identify the appropriate method of solution — justifying your choice from the equation’s classification — and hence find the general solution of each of the following differential equations, in explicit form unless stated otherwise. Where separation requires division by a function of the dependent variable, identify any constant solutions and state whether your final general solution recovers them.

  1. \(t \dfrac{d^3 x}{dt^3} = t^4 +t^{-2}\)

  2. \((1+x)\,dy-x\, dx=0\), where \(y=y(x)\)

  3. \(e^{-x} \dfrac{d^2 y}{dx^2} = 3\)

  4. \(\dfrac{d y}{d x}=\dfrac{e^{x}}{2y}\)

  5. \(\dot{s}+2s=s t^2\), where \(s=s(t)\)

  6. \((e^{2s}-s) \cos r \dfrac{d s}{d r}=e^s \sin 2r\) (you may leave your solution in implicit form)

  7. \((1+x)\,dy-y\, dx=0\), where \(y=y(x)\)

  8. \(\dfrac{d y}{dx}=\dfrac{x}{y} - \dfrac{x}{1+y}\) (you may leave your solution in implicit form)

Remark. Questions 2 and 7 differ by a single letter — yet one is solved by direct integration and the other by separation of variables. Once you have done both, articulate exactly why. This pair is the whole point of classification in miniature.

Initial Value Problems

In the problems below, find an explicit solution for the given initial-value problem. As a challenge, also state the largest interval of definition containing \(x_0\) (or \(t_0\)) on which your solution is valid.

  1. \(\dfrac{\textrm{d} x}{\textrm{d}t}=4(x^2+1); \quad x(\pi/4)=1\)

  2. \(\dfrac{\textrm{d} y}{\textrm{d}x}=\dfrac{y-1}{x-1}; \quad y(2)=2\)

  3. \(\dfrac{\textrm{d}^3 y}{\textrm{d}x^3}=\sin 2x+e^{-2x}+x^2+x+7, \quad \text{subject to} \quad y(0)=0,\; y'(0)=1,\; y''(0)=0\)

Thinking Questions:

  1. In IVP Question 2, dividing by \((y-1)\) discards a constant solution. Which one? Does the initial condition \(y(2)=2\) land on it, or on a member of the general family?
  2. IVP Question 3 needs three conditions where Questions 1 and 2 need only one. State the principle, and connect it to the number of integrations each solution required.

Tutorial 4: Growth Models

Outcomes: At the end of this tutorial you should be able to

- Recognise which growth model (Malthusian, carrying-capacity, or logistic) a verbal description calls for, without needing to solve anything.
- Use problematization - checking a candidate equation's behaviour in obvious special cases - to choose between competing formulations *before* solving.
- Diagnose where a flawed model-building argument actually went wrong, even when the algebra that follows it is executed correctly.
- Construct mathematical growth models from a verbal description, solve the resulting differential equation, and interpret the solution for predictive purposes.

Q1: Recognise the model

Without solving anything, match each scenario below to the model family it calls for: (i) Malthusian (\(\frac{dP}{dt}=kP\)), (ii) carrying-capacity (\(\frac{dP}{dt}=k(M-P)\)), or (iii) logistic (\(\frac{dP}{dt}=kP(M-P)\)).

  1. A colony of bacteria in a nutrient-rich dish, with no space or food constraint for the foreseeable future.

  2. The temperature-like relaxation of a chemical concentration toward a fixed saturation level, where the rate of approach doesn’t depend on how much of the substance is already present — only on how far it still has to go.

  3. A newly introduced population in a reserve, where growth is slow while the population is tiny (few animals to reproduce), fastest around the middle of the reserve’s capacity, and slow again as the reserve fills up.

  4. A rumour spreading through a fixed group of \(N\) people, where the rate new people hear it depends on both how many already know it and how many are still left to tell.

Q2: Problematize before you solve

A conservation group monitors a population of rhino, \(R(t)\), in a reserve with ecological carrying capacity \(K\) (fixed by available grazing). A field ecologist proposes two candidate models: \[\text{(i)}\quad \frac{dR}{dt} = \beta(R-K) \qquad\qquad \text{(ii)}\quad \frac{dR}{dt} = \beta(K-R), \qquad \beta > 0.\]

  1. Without solving either equation, use problematization to determine which candidate is sensible. Justify your choice by checking the sign of \(\frac{dR}{dt}\) in both cases \(R<K\) and \(R>K\).

  2. Even your chosen candidate from (a) has a remaining defect. What does it predict for \(\frac{dR}{dt}\) when \(R\) is very small (close to zero)? Is this realistic for a population of rhino? What refinement to the model would fix this?

Q3: Find the error

A student is asked to model the population \(P(t)\) of an endangered species being reintroduced into a reserve with strict habitat cap \(M\). Their submitted working is reproduced below.

“The population grows at a rate proportional to the difference between the population and the cap, so \[\frac{dP}{dt} = k(P-M), \qquad P(t_0)=P_0, \quad k>0.\] Separating variables and integrating, \[\int \frac{dP}{P-M} = \int k\,dt \;\;\Longrightarrow\;\; \ln|P-M| = kt+c \;\;\Longrightarrow\;\; P(t) = M + (P_0-M)e^{k(t-t_0)}.\] This is the required model.”

  1. The separation and integration steps above are executed correctly. And yet, taking \(P_0=5\), \(M=40\), \(k=0.2\), \(t_0=0\), this solution gives \(P(5) \approx -55\) — a negative population. Explain, in words, exactly where the student’s reasoning went wrong, and at which of the eight modelling steps (Section 2.7) the error was introduced.

  2. Write down the corrected differential equation, and explain — without fully re-solving — why it will not produce this defect.

Q4: Build, solve, and critique

In a particular country the population is expanding at a rate which is directly proportional to the population size \(P\). There is also a net outflow of people from the country (more people are emigrating than immigrating) at a constant rate \(\alpha\).

  1. Before formulating anything, use problematization to check: for very small \(P\) (close to zero), does \(\frac{dP}{dt}=kP-\alpha\) predict sensible behaviour for a real country? What does this suggest about the long-run validity of this model if \(\alpha\) is large relative to \(P_0\)?

  2. If \(t\) is the time, and the population size is \(P_0\) when \(t=0\), derive an expression for the population size \(P\) as a function of \(t\).

  3. Sketch, by inspection, the graph of \(P=P(t)\).

Q5: Build, solve, and interpret

Let \(n=n(t)\) be the size of a population of a certain country at any time \(t\). Suppose the maximum population size the country can sustain is \(N\), and that the rate of growth at any time is proportional to both the current population size and the difference between \(N\) and the current size (constant of proportionality \(\lambda>0\)).

  1. If at \(t=0\), \(n=N_0\), derive a mathematical expression for \(n\) in terms of \(N_0\), \(\lambda\), \(N\), and \(t\).

  2. Sketch, by inspection, the graph of this function.

  3. When is the rate at which the population is increasing at a maximum? Express your answer in terms of \(N_0\), \(\lambda\), and \(N\).

  4. Which of the three named model families from the notes does this equation belong to? What real-world assumption would you need to relax to turn this into a Malthusian model instead?

Q6: A variable rate — and what it means

Suppose \(x_0\) is the number of antelope in a game park at time \(t=0\), and that the population at any subsequent time \(t\geq 0\) is described by \[\frac{dx}{dt} = kx\cos t, \qquad t \geq 0,\] where \(k\) is a constant of proportionality.

  1. Express \(x\) in terms of \(x_0\), \(k\), and \(t\).

  2. Unlike every other model in this tutorial, the growth rate here is not constant — it varies with \(t\). What real biological phenomenon might a factor like \(\cos t\) represent? (Hint: think about what happens twice per cycle, when \(\cos t = 0\).)

  3. Without sketching in detail, describe the long-run behaviour of \(x(t)\) in one sentence: does it grow without bound, settle to an equilibrium, or something else?

Tutorial 5: Decay Models

Outcomes: At the end of this tutorial you should be able to

- Construct a differential equation model for a decaying quantity from a verbal description, and solve it.
- Relate the decay constant to the half-life, and use either one to make quantitative predictions.
- Formulate a decay chain as a system of equations, and use conservation of mass both to shorten the work and to verify a solution.
- Recognise a decay model in disguise, where it is the *difference* from some limiting value that decays rather than the quantity itself.
- Use problematization to identify when a proposed decay model is physically unsound, before solving it.

Q1: A single decaying isotope

The radioactive isotope of lead, Pb-209, decays at a rate proportional to the amount present at time \(t\). Initially, \(1\) gram of the isotope is present.

  1. Write down a differential equation and the initial condition that model the above scenario.
  2. Solve the differential equation found in (a).
  3. Pb-209 has a half-life of \(3{,}3\) hours. Hence find the decay constant \(k\).
  4. How long will it take for \(90\%\) of the lead to decay?
  5. Suppose instead that we had started with \(5\) grams. Without recalculating, state how your answers to (c) and (d) would change, and justify your answer from the form of the solution.

Q2: A decay model in disguise

The growth rate of bacteria in a culture medium is directly proportional to the difference between the number of bacteria and a certain optimal number \(\bar N\).

  1. Derive an expression for the number of bacteria \(n\) as a function of time \(t\), given that the initial number placed in the medium is \(N_0\) and the constant of proportionality is \(\lambda\).
  2. Sketch, by inspection, the graph of \(n = n(t)\) for \(N_0 < \bar N\) and for \(N_0 > \bar N\) on the same axes.
  3. How long does it take for the difference between the number of bacteria and the optimal number to reduce to \(\tfrac{1}{4}\) of the initial difference?
  4. Let \(D(t) = \bar N - n(t)\) denote the gap between the current and optimal populations. Show that \(D\) satisfies \[\frac{dD}{dt} = -\lambda D.\] Hence explain, in a sentence, why a question about bacterial growth belongs in a tutorial on decay.

Q3: A decay chain ending in a stable substance

A radioactive substance \(A\) decays into a stable substance \(B\), with decay constant \(k\). Initially there are \(M\) grams of \(A\) and none of \(B\).

  1. Write down the two differential equations governing \(A\) and \(B\), together with their initial conditions. Explain why the same term \(kA\) appears in both, with opposite signs.
  2. Solve for \(A(t)\) and \(B(t)\). (Hint: once you have \(A(t)\), the equation for \(B\) needs only direct integration.)
  3. Show that \(A(t) + B(t) = M\) for every \(t\). What physical principle does this express, and why is checking it a useful habit?
  4. At what time does the mass of \(B\) first exceed the mass of \(A\)? Express your answer in terms of the half-life, and comment on how neat the result is.

Q4: A two-step chain — reasoning without solving

A radioactive substance \(A\) decays into a radioactive substance \(B\) with decay constant \(k_1\); \(B\) in turn decays into a stable substance \(C\) with decay constant \(k_2\). Initially there are \(M\) grams of \(A\) and none of \(B\) or \(C\).

  1. Write down the three differential equations and their initial conditions.
  2. Without solving anything, explain why \(B(0) = 0\) but \(B(t) > 0\) for small \(t > 0\), and why \(B\) must eventually decrease. Which term in your equation dominates in each phase?
  3. Without solving anything, add your three equations together and hence show that \(A + B + C\) is constant. Why does this mean you never need to integrate the third equation?
  4. The mass of \(B\) reaches a maximum when its gain and loss exactly balance. Write down the condition on \(A\) and \(B\) at that instant.

Q5: A model that fails problematization

A student is asked to model a radioactive substance of initial mass \(M\), and proposes \[\frac{dA}{dt} = -k, \qquad A(0) = M, \qquad k > 0,\] reasoning that “the substance is decaying, so the rate of change should be a negative constant.”

  1. Solve the student’s differential equation.
  2. Use problematization to identify two distinct things that are physically wrong with this model. (One concerns what happens at a particular time; the other concerns the rate itself.)
  3. Compute the “half-life” predicted by the student’s model. How does it differ, structurally, from the half-life of the correct model \(\frac{dA}{dt} = -kA\)?
  4. In one sentence, state the physical assumption the correct model encodes that the student’s model does not.

Tutorial 6: The Method of Undetermined Coefficients

Outcomes: At the end of this tutorial you should be able to

- Explain why an exponential trial solution suits a linear equation with constant coefficients, and write down the auxiliary equation by inspection.
- Use the discriminant of the auxiliary equation to identify which of the three cases applies, and write the complementary function in real form.
- Solve a second-order initial value problem: general solution first, conditions second.
- Decide whether the Method of Undetermined Coefficients applies to a given equation at all, before starting any algebra.
- Choose the form of the ansatz for a given $g(x)$, detect duplication with the complementary function, and adjust the ansatz correctly.
- Find the particular integral, assemble the general solution $y = y_c + y_p$, and only then apply any conditions.

Part A covers Section 3.1.1 (the homogeneous equation) and Part B covers Section 3.2 (the non-homogeneous equation). Several questions ask you to classify, choose a form, or diagnose an error without solving. Those are not warm-ups: they test the decisions that the method depends on, and most lost marks are lost there rather than in the algebra.

Part A: The homogeneous equation

A1: Why an exponential?

For the equation \(ay'' + by' + cy = 0\), with \(a\), \(b\) and \(c\) constants, we try the solution \(y = Ae^{\lambda x}\).

  1. Explain, in your own words, why an exponential is a sensible thing to try. (Think about what the equation is asking of \(y\), \(y'\) and \(y''\).)

  2. Substitute \(y = Ae^{\lambda x}\) into the equation and show that it leads to the auxiliary equation \(a\lambda^2 + b\lambda + c = 0\). At which step do you use the fact that \(e^{\lambda x}\) is never zero?

  3. A classmate suggests that the trial solution \(y = x^n\) would do just as well. Substitute it into \(y'' - 3y' + 2y = 0\) and explain what goes wrong.

A2: Classify before you solve

For each equation below, write down the auxiliary equation, calculate its discriminant, state which of the three cases applies, and write down the general solution. There are no conditions to apply.

  1. \(y'' + 6y' + 9y = 0\)

  2. \(2y'' - 3y' - 2y = 0\)

  3. \(y'' + 2y' + 6y = 0\)

  4. \(y'' + 7y = 0\)

  5. \(y'' - 5y' = 0\)

  6. Without any further calculation, for which of the equations (a)–(e) does every solution tend to zero as \(x \to \infty\)? State, in one sentence, what must be true of the roots of the auxiliary equation for this to happen.

A3: Solve

  1. Find the general solution of \(y'' - 10y' + 25y = 0\).

  2. Solve \(8y'' + 4y' + y = 0\) subject to \(y(0) = 1\) and \(y'(0) = 0\).

  3. Solve \(\ddot{x} + 2\dot{x} - 3x = 0\) subject to \(x(0) = 2\) and \(\dot{x}(0) = -2\).

  4. Find the general solution of \(\dfrac{d^2 S}{dt^2} + 2\dfrac{dS}{dt} + 10S = 0\), and describe in one sentence how \(S\) behaves as \(t\) increases.

A4: When the initial conditions decide the future

This question continues from A3(c), where the general solution of \(\ddot{x} + 2\dot{x} - 3x = 0\) contains one growing exponential and one decaying exponential.

  1. What happens to your solution of A3(c) as \(t \to \infty\)?

  2. Now apply the general initial conditions \(x(0) = x_0\) and \(\dot{x}(0) = v_0\). Show that the coefficient of the growing exponential is \(\tfrac{1}{4}\left(3x_0 + v_0\right)\).

  3. Hence state the condition on \(x_0\) and \(v_0\) under which the solution remains bounded as \(t \to \infty\). Give one pair of initial values that satisfies it, and write down the corresponding solution.

  4. Suppose the initial values are meant to satisfy your condition in (c), but \(v_0\) is measured with a very small error. Describe what the solution does eventually. Why would an engineer be uneasy about a system governed by this equation?

A5: Work backwards

Each of the following is the general solution of a homogeneous, linear, second-order differential equation with constant coefficients. In each case, find the differential equation.

  1. \(y = c_1e^{2x} + c_2e^{-3x}\)

  2. \(y = (c_1 + c_2x)e^{4x}\)

  3. \(y = e^{-x}\left(c_1\cos 2x + c_2\sin 2x\right)\)

  4. Is the differential equation you found in each case the only one with that general solution? Explain.

A6: The suspension of a car

After a car drives over a bump, the vertical displacement \(x(t)\) of its body from the level position (in centimetres, with \(t\) in seconds) is modelled by \[\ddot{x} + b\dot{x} + 16x = 0,\] where the constant \(b \geq 0\) measures how strongly the shock absorbers damp the motion.

  1. For which values of \(b\) does the car body oscillate?

  2. Find the value of \(b\) for which the system is critically damped, and write down the general solution in that case.

  3. The shock absorbers are badly worn, so that \(b = 2\). Write down the general solution, and describe in words what a passenger would feel after the bump.

  4. A mechanic fits shock absorbers that are stiffer than necessary, so that \(b = 10\). Write down the general solution.

  5. Which car returns to level sooner: the one in (b) or the one in (d)? Justify your answer by comparing the exponential terms in the two solutions, without applying any initial conditions. Is “more damping” always better?

A7: Extension — a third-order equation

This question goes one step beyond the second-order equations in the notes. Nothing about the trial solution depends on the order of the equation, and this question asks you to see that for yourself.

Consider \(y''' - 8y = 0\).

  1. Substitute \(y = Ae^{\lambda x}\) and show that the auxiliary equation is \(\lambda^3 - 8 = 0\).

  2. Use the factorisation \(\lambda^3 - 8 = (\lambda - 2)(\lambda^2 + 2\lambda + 4)\) to find all three roots.

  3. Write down the general solution in real form. How many arbitrary constants does it contain, and why is that the number you should expect?

Part B: The non-homogeneous equation

B1: A particular solution and the particular integral

  1. Explain the difference between a particular solution of a differential equation and the particular integral \(y_p\) of a non-homogeneous linear equation. In your answer, say what each one is found from, and whether it must satisfy any initial conditions.

  2. The equation \(y'' + y = 2e^x\) has complementary function \(y_c = c_1\cos x + c_2\sin x\). Consider the three functions \[y_1 = e^x, \qquad y_2 = e^x + 3\cos x, \qquad y_3 = \cos x.\] Which of these are solutions of the non-homogeneous equation? Which of them could be used as the particular integral \(y_p\)?

  3. What does your answer to (b) tell you about whether the particular integral is unique? Does the general solution \(y = y_c + y_p\) depend on which one you choose?

  4. Which of \(y_1\), \(y_2\) and \(y_3\), if any, is the particular solution of the initial value problem \(y'' + y = 2e^x\), \(y(0) = 1\), \(y'(0) = 1\)?

B2: Does the method apply?

Without solving anything, decide whether the Method of Undetermined Coefficients can be used to find a particular integral of each equation below. If it cannot, give the reason. If it can, write down the form of the ansatz you would use, after checking for duplication.

  1. \(y'' + y = \tan x\)

  2. \(xy'' + y = e^x\)

  3. \(y'' + 4y = x\cos 2x\)

  4. \(y' - 3y = e^{3x}\)

  5. \(y'' + y' = \ln x\)

  6. \(yy'' + y = x\)

B3: Choose the ansatz

Consider the equation \(y'' - 4y' + 4y = g(x)\).

  1. Find the complementary function \(y_c\).

  2. For each \(g(x)\) below, write down the form of the ansatz for \(y_p\). Do not find the coefficients.

    1. \(g(x) = \sin 2x - x^2\)

    2. \(g(x) = xe^{2x}\)

    3. \(g(x) = 5e^{2x}\)

    4. \(g(x) = e^{2x}\cos x\)

  3. In which of (i)–(iv) did you have to adjust your first guess, and by what power of \(x\)? In (iv) the factor \(e^{2x}\) appears in both \(g(x)\) and \(y_c\). Explain why that, on its own, does or does not amount to duplication.

B4: Find the error

A student is asked to find the general solution of \(y'' - y' - 2y = 3e^{2x}\). Their submitted working is reproduced below.

“The auxiliary equation is \(\lambda^2 - \lambda - 2 = 0\), so \(\lambda = 2\) or \(\lambda = -1\), and \[y_c = c_1e^{2x} + c_2e^{-x}.\] Since \(g(x) = 3e^{2x}\), I try \(y_p = Ae^{2x}\). Then \(y_p' = 2Ae^{2x}\) and \(y_p'' = 4Ae^{2x}\). Substituting, \[4Ae^{2x} - 2Ae^{2x} - 2Ae^{2x} = 3e^{2x} \;\;\Longrightarrow\;\; 0 = 3e^{2x}.\] This is impossible, so the equation has no particular integral.”

  1. Every line of the student’s algebra is correct. Which step of the method did they leave out?

  2. Explain, without calculating anything, why \(Ae^{2x}\) could never have produced \(3e^{2x}\) on the right-hand side, whatever the value of \(A\).

  3. Write down the correct ansatz, find \(y_p\), and give the general solution.

  4. Suppose the right-hand side had been \(3e^{-x}\) instead, or \(3e^{3x}\). For each, state whether the first guess would need adjusting, and why.

B5: Solve

  1. Find the general solution of \(y'' - 10y' + 25y = 30x + 38\). (You already have \(y_c\) from A3(a).)

  2. Find the general solution of \(y'' + 3y = -48x^2e^{3x}\).

  3. Solve \(y'' - y' - 2y = 4x + 3e^{2x}\) subject to \(y(0) = 1\) and \(y'(0) = 0\). (Use the superposition principle, Theorem ??: treat \(4x\) and \(3e^{2x}\) separately. You have already done half of the work in B4.)

  4. Consider \(y'' + 3y = 6x\) subject to \(y(0) + y'(0) = 0\) and \(y(1) = 0\).

    1. Is this an initial value problem? Look carefully at where each condition is imposed.

    2. Find the solution. The constants do not come out neatly: leave them in exact form, in terms of \(\cos\sqrt{3}\) and \(\sin\sqrt{3}\).

    3. At what stage did you apply the two conditions, and why would applying them to \(y_c\) alone have given the wrong answer?

B6: Driving a system at its own frequency

An undamped system is at rest when a periodic driving force is switched on. Its displacement \(y(x)\) satisfies \[y'' + 9y = 6\sin\omega x, \qquad y(0) = 0, \quad y'(0) = 0,\] where \(\omega\) is the frequency of the driving force.

  1. Write down the complementary function. What is the natural frequency of the system?

  2. Take \(\omega = 2\). Solve the initial value problem. Does the solution remain bounded?

  3. Now take \(\omega = 3\). Explain why the ansatz you used in (b) fails, and solve the initial value problem with the correct one.

  4. Describe how your solution in (c) behaves as \(x\) increases, and sketch it. What is this phenomenon called?

  5. Without solving, consider the damped version \(y'' + 2y' + 9y = 6\sin 3x\). Does the ansatz \(A\cos 3x + B\sin 3x\) duplicate any term of the complementary function now? What does this suggest about why real structures, which always have some damping, usually survive being driven at their natural frequency?

Tutorial 7: Newton’s Law of Cooling and Oscillatory Models

Outcomes: At the end of this tutorial you should be able to

- Use problematization to choose the correct sign in Newton's Law of Cooling and Warming, and recognise a model that cannot be right.
- Formulate and solve a cooling or warming problem, find the constant $k$ from data, and use the solution to make predictions forwards and backwards in time.
- Use the fact that the temperature difference decays exponentially to reason quickly, sometimes without a calculator.
- Classify a free spring or circuit as undamped, underdamped, critically damped or overdamped from its auxiliary equation, and describe the motion in words.
- Formulate a spring--mass or RLC model from a description, solve the initial value problem, and interpret the solution.
- Explain resonance in terms of the natural frequency, and recognise when it occurs.

Part A covers Newton’s Law of Cooling and Warming (Section 2.10) and Part B covers oscillatory models (Section 3.3). Throughout, use the convention \(\dfrac{dT}{dt} = -k(T - T_m)\) with \(k > 0\), and remember the order of work for second-order problems: general solution first, conditions second.

Part A: Newton’s Law of Cooling and Warming

A1: Which model?

A pie is taken out of an oven and left on a counter in a kitchen at \(22^{\circ}\)C. Three students propose models for its temperature \(T(t)\), each with \(k > 0\): \[\text{(i)}\;\; \frac{dT}{dt} = k(T - 22) \qquad\quad \text{(ii)}\;\; \frac{dT}{dt} = -kT \qquad\quad \text{(iii)}\;\; \frac{dT}{dt} = -k(T - 22)\]

  1. Without solving anything, use problematization to explain why model (i) cannot be correct. (Check the sign of \(\frac{dT}{dt}\) when the pie is hotter than the kitchen.)

  2. What temperature does model (ii) predict the pie approaches in the long run? Why is this unrealistic?

  3. Show that model (iii) passes both of the checks you used in (a) and (b). Would it still behave sensibly if the “pie” were a tub of ice cream at \(-15^{\circ}\)C taken out of a freezer? Explain.

A2: A pot of pap

On a winter evening, a pot of pap at \(85^{\circ}\)C is taken off the stove and left in a kitchen at \(15^{\circ}\)C. Fifteen minutes later it is at \(50^{\circ}\)C.

  1. Write down the differential equation and initial condition for the temperature \(T(t)\) of the pap, \(t\) minutes after it is taken off the stove. State what each symbol represents.

  2. Solve your differential equation.

  3. Find the exact value of \(k\).

  4. Without a calculator, find the temperature of the pap after \(45\) minutes. (What happens to the difference \(T - 15\) every fifteen minutes?)

  5. How long after it is taken off the stove will the pap be at \(20^{\circ}\)C?

A3: Two cold drinks

Two identical cans of cold drink, each at \(5^{\circ}\)C, are taken out of a fridge at the same moment. One is left on a sunny stoep at \(30^{\circ}\)C; the other is left in a kitchen at \(20^{\circ}\)C. Assume that both cans warm according to Newton’s law with the same value of \(k\).

  1. Without solving, which can warms faster at the moment it is taken out? Justify your answer from the differential equation.

  2. Sketch the temperatures of both cans against time on the same axes, marking each room temperature.

  3. Show that, for both cans, the temperature difference from the surroundings halves every \(\dfrac{\ln 2}{k}\) minutes.

  4. Which can reaches \(15^{\circ}\)C first? Answer this without knowing the value of \(k\).

A4: When was the coffee poured?

At 09:00 a lecturer finds a forgotten cup of coffee on a desk. Its temperature is \(40^{\circ}\)C, and the office is at a constant \(20^{\circ}\)C. At 09:10 the coffee is at \(35^{\circ}\)C. Freshly made coffee from the office machine is at \(85^{\circ}\)C.

  1. Taking \(t = 0\) at 09:00, find \(T(t)\) and the value of \(k\).

  2. At what time was the coffee poured? Explain why your value of \(t\) is negative.

  3. State two assumptions you made, and for each one say whether it is likely to make your estimate of the time too early or too late.

A5: A model that fails

A student uses Newton’s law to model an ice cube, initially at \(-10^{\circ}\)C, placed on a plate in a room at \(25^{\circ}\)C. Their model predicts that the ice cube’s temperature rises smoothly past \(0^{\circ}\)C and reaches \(20^{\circ}\)C after a few minutes.

  1. Describe what actually happens to the temperature of the ice cube once it reaches \(0^{\circ}\)C.

  2. Which assumption of Newton’s law fails during that stage, and why?

  3. After the ice has completely melted, could Newton’s law be a reasonable model again? Explain.

Part B: Oscillatory models

B1: Read the motion from the equation

Each equation below models the free motion of a mass on a spring, with \(x\) the displacement from equilibrium. For each one, find the discriminant of the auxiliary equation, name the type of motion (undamped, underdamped, critically damped or overdamped), and sketch the typical shape of \(x(t)\) for a mass pulled aside and released from rest. Do not find the solutions.

  1. \(\ddot{x} + 16x = 0\)

  2. \(\ddot{x} + 6\dot{x} + 5x = 0\)

  3. \(\ddot{x} + 4\dot{x} + 4x = 0\)

  4. \(2\ddot{x} + \dot{x} + 8x = 0\)

  5. \(\ddot{x} - \dot{x} + 4x = 0\). This equation cannot describe a real mass on a spring with friction. Why not? What would its solutions do?

B2: Build it, then solve it

A mass of \(0{,}5\) kg hangs on a spring with spring constant \(k = 8\) N/m, and a damper provides a resistance with damping constant \(b = 4\) N\(\cdot\)s/m. The mass is pulled \(0{,}1\) m below its equilibrium position and released from rest.

  1. Write down the initial value problem for the displacement \(x(t)\), taking downwards as positive.

  2. Which type of damping is this? Show your working.

  3. Solve the initial value problem.

  4. Does the mass ever pass through its equilibrium position? Justify your answer from your solution.

B3: Designing a car’s suspension

A car of mass \(m = 1000\) kg rests on springs with total spring constant \(k = 25\,000\) N/m. Its vertical motion after a bump is modelled by \(m\ddot{x} + b\dot{x} + kx = 0\), where \(b\) is the damping constant of the shock absorbers.

  1. Without shock absorbers (\(b = 0\)), find the natural frequency and the period of the car’s bounce.

  2. Find the damping constant that makes the suspension critically damped.

  3. Over time, the shock absorbers wear, so that \(b = 8000\) N\(\cdot\)s/m. Find the roots of the auxiliary equation and describe how the car now moves after a bump.

  4. Taking \(x(0) = 0{,}05\) m and \(\dot{x}(0) = 0\), solve the initial value problem in (c).

  5. Why would an engineer choose a damping constant close to, but not far above, the value in (b)?

B4: The same equation in a circuit

An RLC circuit with no voltage source has inductance \(L = 1\) H, capacitance \(C = \tfrac{1}{26}\) F and resistance \(R\).

  1. Write down the differential equation for the charge \(q(t)\) on the capacitor, and state which quantity in the spring–mass model each of \(L\), \(R\) and \(\tfrac{1}{C}\) corresponds to.

  2. Find the resistance for which the circuit is critically damped.

  3. Take \(R = 2\ \Omega\). The capacitor is initially uncharged, and the initial current is \(\dot{q}(0) = 10\) A. Find \(q(t)\), and describe the behaviour of the charge.

B5: Pushing at the right frequency

A machine part of mass \(1\) kg, mounted on a spring with \(k = 36\) N/m and negligible damping, is shaken by a motor with force \(\cos\omega t\): \[\ddot{x} + 36x = \cos\omega t.\]

  1. What is the natural frequency of the part? For which value of \(\omega\) does resonance occur?

  2. For \(\omega \neq 6\), a particular integral is \(x_p = \dfrac{\cos\omega t}{36 - \omega^{2}}\). Verify this by substitution.

  3. Compare the amplitude of \(x_p\) when \(\omega = 3\) and when \(\omega = 5{,}9\). What does this tell you about running a motor at a frequency close to the natural frequency, even if it is not exactly equal?

  4. Real machine parts always have some damping. Explain, using the complementary function, why damping prevents the amplitude from growing without bound at \(\omega = 6\).

B6: Find the error

A student is asked to solve \(\ddot{x} + 2\dot{x} + 17x = 0\), with \(x(0) = 1\) and \(\dot{x}(0) = 3\). Their working is reproduced below.

“The auxiliary equation \(\lambda^2 + 2\lambda + 17 = 0\) gives \(\lambda = -1 \pm 4i\), so \[x = e^{-t}\left(c_1\cos 4t + c_2\sin 4t\right).\] From \(x(0) = 1\), \(c_1 = 1\). Differentiating, \(\dot{x} = e^{-t}\left(-4c_1\sin 4t + 4c_2\cos 4t\right)\), so \(\dot{x}(0) = 4c_2 = 3\) and \(c_2 = \tfrac34\). Therefore \[x = e^{-t}\left(\cos 4t + \tfrac34\sin 4t\right)."\]

  1. Check the student’s answer against the second initial condition. Does it satisfy \(\dot{x}(0) = 3\)?

  2. Identify the error, and explain why it is easy to make.

  3. Find the correct solution.