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?