MTH2032 Differential Equations with Modelling
The Complete Exam Bible · S2 2026MTH2032 Overview
- Monash University
- Semester 2 2026
- Level 2 mathematics
- Twelve teaching weeks
- ODEs then PDEs
What MTH2032 teaches
Differential equations relate rates of change, and MTH2032 teaches how to interpret, analyse and solve them, with a strong emphasis on modelling physical systems. The unit is split into two halves.
- Assessed by Continuous tasks worth 50% and a cumulative final examination worth 50%.
- Final emphasis The final leans towards the partial differential equations half of the unit.
- Hardest step Choosing and justifying the method before any calculation starts.
- How to prepare Write every solution as classify, justify, work and conclude.
What you get in the full MTH2032 Exam Bible
- 12 chapters, 44 pages — every examinable topic, week by week
- 33 original practice questions, each with a model answer in the Final Examination format
- 12 worked examples, solved the way MTH2032 marks
- 36 computed figures, redrawn from the course’s own numbers
- Exam-morning plan for the Final Examination (50%) — what to revise last, where marks are lost
Free on this page: assessment table, chapter map, one worked example, key terms, FAQ. The rest is in the full exam bible.
How MTH2032 is assessed
| Component | Weight | Format |
|---|---|---|
| Participating and engaging in applied classes | 4% | Weekly; banded from participation marks out of 11 |
| Own-time and Wrap-up quizzes | 6% | Weekly; 6% for a total quiz mark of at least 75%, 3% from 50% |
| Quizzes in applied classes | 10% | In Weeks 2, 6, 9 and 11; 15 minutes each, 2.5% each |
| Assignment Verification Tasks | 10% | In Weeks 4 and 8; supervised, closed-book, 30 minutes, 5% each |
| Midterm Tests | 20% | In Weeks 5 and 10; 60 minutes, 10% each, one handwritten A4 page, one side |
| Final Examination | 50% | In the examination period; cumulative, weighted towards PDEs; one handwritten A4 page, both sides |
Weights come from the Semester 2 2026 unit information on Moodle and total 100%. The Monash handbook groups the same assessment as Demonstration 50% and Examination 50%. Confirm dates and the final examination details on Moodle.
Assessment structure
Bar widths are proportional to the six published weights, which sum to 100%.
What other students are asking about MTH2032
- Where can I find lecture notes or study guides for MTH2032 at Monash?
- The MTH2032 study guide in AskSia Library covers Differential Equations with Modelling in 12 chapters, with the assessment table, a chapter-by-chapter topic map, key terms and a worked example, free to read on this page.
- Is there a final exam in MTH2032, and how much is it worth?
- Yes. In the offering this AskSia Library guide is built from, Final Examination carries 50% of the MTH2032 mark; the full split is Participating and engaging in applied classes 4%, Own-time and Wrap-up quizzes 6%, Quizzes in applied classes 10%, Assignment Verification Tasks 10%, Midterm Tests 20%, Final Examination 50%.
- What does MTH2032 cover on modelling with first-order odes and ivps?
- In AskSia Library's MTH2032 guide, the Modelling with First-Order ODEs and IVPs chapter explains balance laws over a short time step, Newton's law of cooling, existence and uniqueness, and recasting higher-order equations as systems.
- What does MTH2032 cover on general theory of second-order linear odes?
- In AskSia Library's MTH2032 guide, the General Theory of Second-Order Linear ODEs chapter explains superposition, bases of solutions, the Wronskian test and characteristic roots written with cosh and sinh.
- What does MTH2032 cover on free oscillations and particular solutions?
- In AskSia Library's MTH2032 guide, the Free Oscillations and Particular Solutions chapter explains mass-spring models, under, critical and over damping, undetermined coefficients with the modification rule, and variation of parameter.
Current MTH2032 dates
| Date | Item | Control |
|---|---|---|
| 9 November 2026 | Final exam consultation | Listed on the unit page as Monday 9 November, 1pm to 2:30pm; the examination itself falls in the examination period. |
Dates are as published in the Semester 2 2026 Moodle unit page. Confirm exact deadlines and submission settings in the live LMS.
What MTH2032 covers
The chapters follow the twelve teaching weeks of the 2026 lecture notes. Part I builds first-order and second-order ODE methods and their models; Part II classifies PDEs and solves them by characteristics, separation of variables and Fourier series.
First-Order ODEs: Separable, Linear, Exact and Homogeneous
Type recognition, variation of parameter, the exactness test and the ratio substitutionWeek 1 introduces the four first-order families that come with a guaranteed solution procedure: separable, linear, exact and homogeneous type. The skill being trained is recognition.
- Recognising separable equations
- Standard form of a linear equation
- Variation of parameter and formula (2.2)
- The cross-derivative test for exactness
- Finding a potential function
- The substitution y = vx for ratio equations
Modelling with First-Order ODEs and IVPs
Balance laws over a short time step, Newton's law of cooling, existence and uniqueness, and recasting higher-order equations as systemsWeek 2 turns physical rules into differential equations and asks when an initial value problem has exactly one solution.
- Balance between t and t + δt
- Radioactive decay and half-life
- Newton's law of cooling
- Theorem 5.2 for scalar equations
- Continuity checks without limits
- Recasting to first-order systems
General Theory of Second-Order Linear ODEs
Superposition, bases of solutions, the Wronskian test and characteristic roots written with cosh and sinhWeek 3 treats second-order linear equations as linear algebra. Superposition makes the solution set of a homogeneous equation a space, the Basis theorem shows that two independent solutions span it, and the Wronskian decides independence.
- Standard form and Theorem 7.1
- General solution y_h + y_p
- Superposition principle
- Basis theorem
- Wronskian theorem
- Characteristic roots and Table 1
Free Oscillations and Particular Solutions
Mass-spring models, under, critical and over damping, undetermined coefficients with the modification rule, and variation of parameterWeek 4 applies the constant-coefficient theory to a mass on a spring and then moves to non-homogeneous equations. Free oscillations come in three damping regimes decided by comparing c² with 4km.
- Hooke's law and my″ + ky = 0
- Natural frequency and period
- Under, critical and over damping
- Dimensional analysis of k and c
- Trial functions and the modification rule
- Variation of parameter for second order
Forced Oscillations, BVPs and Reduction of Order
Beats and resonance, damped steady states, Dirichlet, Neumann and mixed boundary value problems, eigenvalue problems and reduction of orderWeek 5 adds an external force to the spring and then changes the kind of problem being solved.
- Undamped forcing and beats
- Resonance and the factor t
- Damped steady state and transient
- Dirichlet, Neumann and mixed BVPs
- The eigenvalue problem X″ = KX
- Reduction of order and missing variables
Classifying PDEs and Deriving Physical Models
Order, grades of linearity, homogeneity and type, and how the advection, wave, heat and Laplace equations are derivedWeek 6 opens Part II with vocabulary and models. A PDE is classified by order, by grade of linearity from linear through semilinear and quasilinear to fully nonlinear, by homogeneity when it is linear, and by type when it is second-order semilinear.
- Order of a PDE
- Linear, semilinear, quasilinear, fully nonlinear
- Homogeneity and superposition
- Hyperbolic, parabolic and elliptic type
- Arbitrary functions in general solutions
- Deriving the advection, wave, heat and Laplace equations
Characteristics for First-Order PDEs
Semilinear cases, canonical forms and the geometric method for quasilinear equationsWeek 7 solves first-order PDEs by following their characteristic curves. For a semilinear equation the characteristic curves solve dy/dx = B/A, and changing to coordinates that follow them turns the PDE into an ODE along each curve, the canonical form.
- Standard form A u_x + B u_y = C
- Characteristic ODE dy/dx = B/A
- Choosing s and τ
- Canonical form and inversion
- Characteristic equations in (x, y, u)
- Intersecting surfaces approach
Cauchy Problems and Domains of Definition
Solving from data on a curve, the non-characteristic condition, the Jacobian test, blow-up and crossing characteristicsWeek 8 applies characteristics to Cauchy problems, where the solution is prescribed along a curve rather than at a point. The data can be fitted to a general solution or used as an initial curve from which characteristics are swept out.
- Cauchy surface and Cauchy data
- Method 1: fitting the arbitrary function
- Method 2: sweeping out the initial curve
- Cauchy-Kovalevskaya theorem
- Non-characteristic condition and the Jacobian
- Bounded data, blow-up and crossing characteristics
Separation of Variables with Homogeneous Boundaries
Product solutions for the wave, heat and Laplace equations, spatial eigenvalue problems and superpositionWeek 9 introduces separation of variables for the wave, heat and Laplace equations when two boundary conditions are homogeneous and the remaining data are already sines or cosines.
- When separation of variables applies
- Product solutions and the separation constant
- Spatial eigenvalue problem
- Time factors for wave and heat
- Laplace equation on a rectangle
- Superposition with boundary conditions
Fourier Series and Half-Range Expansions
Periodic functions, Euler formulae, the representation theorem, convergence rates and even or odd periodic extensionsWeek 10 develops Fourier series, the tool that lets separation of variables handle general data.
- Periodic functions and fundamental period
- Euler formulae for the coefficients
- Representation theorem and jump values
- Convergence rate and smoothness
- Even and odd functions
- Half-range cosine and sine series
Separation of Variables with Non-Periodic Conditions
Fourier coefficients for initial data, steady state plus transient for boundary data, and splitting Laplace problemsWeek 11 completes separation of variables. When initial data are not already sines or cosines, they are expanded in the eigenfunctions with the Euler formulae, and wave problems with two non-zero initial conditions need two sets of coefficients.
- Fourier coefficients for initial data
- Two inhomogeneous initial conditions
- Steady state and transient
- Theorem 28.3 for non-zero boundaries
- Splitting Laplace problems as u_v + u_h
- Checking the boundary of the sum
Characteristics for Second-Order Semilinear PDEs
Factorising the operator, the auxiliary equation, hyperbolic and parabolic canonical forms, and d'Alembert's solutionWeek 12 returns to characteristics for second-order semilinear PDEs.
- Factorising the second-order operator
- Auxiliary equation and its sign
- Hyperbolic canonical form u_vw
- Parabolic canonical form u_ww
- Elliptic equations and Poisson form
- D'Alembert's solution
Part I treats ordinary differential equations: the separable, linear, exact and homogeneous-type first-order families, models built from balance laws, existence and uniqueness theorems, the general theory of second-order linear equations, mass-spring oscillations with and without damping or forcing, boundary value problems and reduction of order.
Part II treats partial differential equations: classification by order, linearity and type, derivation of the advection, wave, heat and Laplace equations, the method of characteristics, Cauchy problems and their domains of definition, separation of variables, Fourier series and d'Alembert's solution.
How the unit is organised
Twelve teaching weeks follow the 2026 lecture notes chapter by chapter.
Weeks 1 to 5 cover Part I and Weeks 6 to 12 cover Part II, with each week combining seminars, a two-hour applied class and own-time and wrap-up quizzes.
Released problems for the two supervised Assignment Verification Tasks appear two weeks before each task, and the questions asked in the room may change parameters or conditions, so understanding the method matters more than remembering an answer.
What the assessment rewards
Half of the final mark comes from continuous work: participation, weekly quizzes, quizzes in applied classes, the verification tasks and two midterm tests.
The other half is a cumulative final examination with an emphasis on the PDE material. The unit expects proper mathematical communication, which means full sentences that explain why each step is valid, not just a chain of equations.
Worked example · free
Separate a heat problem with a single sine mode
- 1Assume u = X(x)T(t) and separate: T′/T = X″/X = K, with X(0) = X(π) = 0 from the boundary conditions.
- 1The eigenvalue problem gives non-zero X only for K = −n², with X_n = sin nx and n = 1, 2, 3 and so on.
- 1The time equation T′ = −n²T gives T_n = e^(−n²t).
- 1Superpose u = Σ b_n sin(nx) e^(−n²t); the initial data contain only n = 2, so b_2 = 6.
- 1Conclude u = 6 sin(2x) e^(−4t) and check both boundary values and the initial value.
Key terms
- Initial Value Problem
- An ODE together with values of the solution, and of enough derivatives, at one starting point.
- Exact Equation
- A first-order equation M dx + N dy = 0 whose left side is the differential of a single potential function u.
- Wronskian
- The determinant y1y2′ − y2y1′ that is never zero exactly when two solutions are linearly independent.
- Characteristic Equation
- The polynomial equation in λ obtained by substituting e^(λx) into a constant-coefficient linear ODE.
- Resonance
- Unbounded growth of an undamped oscillator driven at its own natural frequency.
- Eigenvalue Problem
- A boundary value problem asking for the constants K that allow a non-zero solution of X″ = KX.
- Quasilinear PDE
- A PDE whose highest derivatives appear linearly with coefficients that may depend on the solution and its lower derivatives.
- Characteristic Curve
- A curve along which a first-order PDE reduces to an ordinary differential equation.
- Cauchy Data
- The values prescribed for the solution along a given curve in a Cauchy problem.
- Domain of Definition
- The region in which the solution of a Cauchy problem is defined and unique.
- Steady State
- A time-independent solution of an evolution equation that satisfies the given boundary conditions.
- Fourier Series
- A representation of a periodic function as a constant plus a sum of cosines and sines sharing one period.
- Half-Range Expansion
- A cosine or sine series built from the even or odd periodic extension of a function defined on a finite interval.
- Canonical Form
- The simplified form of a PDE obtained after changing to coordinates aligned with its characteristics.
MTH2032 FAQ
How is the unit assessed?
The Semester 2 2026 summary lists six weighted tasks: participation in applied classes 4%, own-time and wrap-up quizzes 6%, four quizzes in applied classes 10%, two Assignment Verification Tasks 10%, two midterm tests 20%, and the final examination 50%. The Monash handbook groups the first five as Demonstration at 50% and the final as Examination at 50%.
What does the final examination cover?
The unit information describes the final examination as cumulative while placing more weight on the partial differential equations half. Characteristics, Cauchy problems, separation of variables, Fourier series and second-order canonical forms therefore deserve the largest share of revision, while the ODE theory from Part I still underpins many PDE steps such as eigenvalue problems.
Can I bring notes into the exams?
For the final examination the unit allows a single A4 page of handwritten notes written on both sides. For each midterm test it allows a single A4 page of handwritten notes on one side only. The unit says further details of the final are released later in the semester, so confirm the current arrangements on Moodle shortly before the examination period.
How do the Assignment Verification Tasks work?
Problems are released two weeks ahead. The task itself is supervised, closed-book and takes thirty minutes in your applied class, and its questions are similar to the released ones but may change parameters, assumptions or conditions. Each task is worth 5%. Generative AI may be used while preparing, provided you check its output against the lecture notes.
When can I use separation of variables on a PDE?
The lecture notes recommend trying it when the PDE is linear, homogeneous and at least second order, with homogeneous boundary conditions on a pair of opposite boundaries. Non-zero boundary values for heat problems are first removed by subtracting a steady state, and Laplace problems with data on every side are split into two pieces that each have two homogeneous edges.
Why does the unit write solutions with cosh and sinh?
When the characteristic equation has distinct real roots, the course table writes the basis as an exponential times cosh and sinh rather than two separate exponentials. Both span the same solutions, but cosh equals one and sinh equals zero at the origin, so a zero boundary or initial value removes one constant immediately, which saves work in boundary value and eigenvalue problems.
What makes a Cauchy problem answer complete?
A Cauchy problem is not finished until you state its domain of definition: the region reached by characteristics that cross the data curve exactly once. The domain can end at the ends of the data, where data run along a characteristic, where the solution blows up along a characteristic, or where characteristic projections cross each other.
Questions students ask in each MTH2032 chapter
- How do I tell which first-order method to use?
- Ask the questions in a fixed order: can the right-hand side split into a function of x times a function of y; can the equation be rearranged as y′ + py = q with coefficients in x; does M_y equal N_x; does everything depend on y/x. The first yes chooses the method, although some equations fit two families and either route works. Chapter 01 · First-Order ODEs: Separable, Linear, Exact and Homogeneous
- How do I show that an initial value problem has a unique solution?
- Write the equation as y′ = f(x, y), argue that f and its partial derivative in y are continuous on a rectangle, check that the starting point lies inside that rectangle, and then quote the fundamental existence-uniqueness theorem with its conclusion on an interval around the starting point. Each of those steps carries marks in the problem-set solutions. Chapter 02 · Modelling with First-Order ODEs and IVPs
- What does the Wronskian theorem let me conclude?
- For two solutions of a homogeneous equation with continuous coefficients on an interval, the Wronskian is either never zero, in which case the solutions are independent, or identically zero. So a pair whose Wronskian vanishes at one point but not at another cannot both solve such an equation, which is the argument behind several released verification problems. Chapter 03 · General Theory of Second-Order Linear ODEs
- How do I decide which damping case I am in?
- Compare c² with 4km from the equation my″ + cy′ + ky = 0. If c² is smaller the roots are complex and the motion oscillates in a decaying envelope; if they are equal there is a repeated root and critical damping; if c² is larger the roots are real and the motion is over-damped, written with cosh and sinh. Chapter 04 · Free Oscillations and Particular Solutions
- Why does resonance not occur when there is damping?
- With damping the characteristic roots have a non-zero real part, so neither cosine nor sine at the driving frequency solves the homogeneous equation. The ordinary trial works without the factor t, and the response stays bounded. The notes make this point when they treat damped forcing as a single case. Chapter 05 · Forced Oscillations, BVPs and Reduction of Order
- How do I find the most specific linearity class?
- Start from the highest-order terms. If they are linear with coefficients depending only on the independent variables, check the remaining terms: all linear means linear, otherwise semilinear. If the highest-order coefficients involve u or lower derivatives, the equation is quasilinear. If the highest derivatives appear non-linearly, for example squared, it is fully nonlinear. Chapter 06 · Classifying PDEs and Deriving Physical Models
- How do I choose the second coordinate τ?
- Any differentiable function works if its level curves are never parallel to the characteristics and each characteristic meets each level curve exactly once, so that the change of variables can be inverted. Taking τ = x or τ = y is usually enough; say in your answer why the choice is acceptable. Chapter 07 · Characteristics for First-Order PDEs
- Which method should I use for a Cauchy problem?
- If the general solution is easy to find, fit the data to its arbitrary function, as with constants in Part I. Otherwise parametrise the data as an initial curve at τ = 0 and solve the characteristic equations from each point. The second method also produces the Jacobian, which helps locate the edge of the domain of definition. Chapter 08 · Cauchy Problems and Domains of Definition
- Why can I set both sides of the separated equation equal to one constant?
- After separation the left side depends only on t and the right side only on x, yet they are equal for every x and t. Changing t alone cannot change the right side, so the left side must be constant, and likewise the other way round. That shared constant is the separation constant K. Chapter 09 · Separation of Variables with Homogeneous Boundaries
- What value does a Fourier series take at a jump?
- By the representation theorem, at a point where f is discontinuous the sum of the series is the mean of the two one-sided limits. Remember that periodicity matters at the ends of a period: the left limit at the start of a period is the value approached at the end of the previous one. Chapter 10 · Fourier Series and Half-Range Expansions
- How do I handle a heat problem whose ends are held at different temperatures?
- Find the steady state first: for the plain heat equation it is the straight line joining the two end temperatures. Subtract it from u. The remainder satisfies the heat equation with zero boundary values and a shifted initial profile, so ordinary separation of variables applies. Add the steady state back at the end. Chapter 11 · Separation of Variables with Non-Periodic Conditions
- Why is there a minus sign in the auxiliary equation?
- Expanding the factorised operator gives λ1 + λ2 = B/A and λ1λ2 = C/A, so the roots solve Aλ² − Bλ + C = 0. The notes warn that the minus sign is the part most often forgotten. With the wrong sign the characteristic slopes are reversed and the final check fails. Chapter 12 · Characteristics for Second-Order Semilinear PDEs
How to study for the exam
Revise in the order the final examination weights the unit: Part II first, then Part I as the toolkit it depends on.
Start with separation of variables, because it combines most of the course: classification, the three-case eigenvalue problem from Week 5, Fourier coefficients from Week 10 and the steady-state shift from Week 11. Then work through characteristics and Cauchy problems, practising the domain of definition until you can name what bounds it without a sketch.
For Part I, make sure the characteristic-root table, the modification rule and variation of parameter are automatic, since they reappear inside PDE solutions. For every practice item, write the four-line frame used throughout this guide: classify, justify the method, work one step per line, conclude with the solution set or region.
Build your double-sided notes sheet as you go, and test it by solving a mixed set of problems using only the sheet.
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