Vu's Maths Hub: HSC Differential Equations: Modelling, Slope Fields, and Separable ODEs

Intro

The HSC Differential Equations booklet is a free Mathematics Extension 1 and Extension 2 resource with 57 worked problems covering exponential growth and decay, resistive motion, simple harmonic motion, slope fields, separable equations, and the logistic model. It is written for Extension 1 and 2 students, motivated learners, and teachers wanting concise reference plus enrichment and designed for structured HSC revision on Vu's Maths Hub.

This deep-dive introduces HSC Differential Equations — browser-readable, aligned with the NESA syllabus.

Summary

The HSC Differential Equations booklet offers a fundamentals review, 57 tiered problems with solutions, appendices, and a conclusion that distils exam habits. Open HSC Differential Equations — no account required. Use this post to plan how to work through a 117-page booklet efficiently.

What is this booklet?

This booklet is written for Extension 1 and 2 students, motivated learners, and teachers wanting concise reference plus enrichment.

Focus: exponential growth and decay, resistive motion, simple harmonic motion, slope fields, separable equations, and the logistic model.

Topics covered:

Exponential growth and decay
Resistive motion and terminal velocity
Escape velocity and gravitation
Simple harmonic motion
Water tanks and Torricelli models
Slope fields
Separable equations
Logistic model

How to use it:

Review fundamentals before Part 1
Try each problem without hints, then compare solutions
Study appendices A–C for exam essentials; D–H for enrichment
Prerequisites: derivatives, integrals, basic physics context

Approximately 117 pages, CC BY 4.0, readable at HSC Differential Equations.

Key fundamental reviews

The booklet opens with a dedicated Fundamentals Review — read it before Part 1. It is a compact reference for notation, formulas, and reasoning patterns:

What a differential equation is and what a solution represents
Separable equations and integration techniques
Exponential growth/decay models
Slope fields and reading qualitative behaviour
Initial conditions and particular solutions
Autonomous equations and equilibrium ideas

Why fundamentals matter

What a differential equation is and what a solution represents — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Separable equations and integration techniques — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Exponential growth/decay models — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Slope fields and reading qualitative behaviour — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Initial conditions and particular solutions — appears across multiple problem tiers; redo the fundamentals example, then attempt two Part 1 questions that use it.

Students who skip this section often repeat the same algebra errors in Part 2. Treat HSC Differential Equations fundamentals as a closed-book quiz first.

Problems and how to use them

The HSC Differential Equations booklet packs 57 practice problems into roughly 117 pages — well beyond a single textbook chapter. Each item includes worked solutions; many include Takeaways that highlight the method to reuse in exams.

Overall structure

Problems: warm-up, medium, and advanced modelling problems with detailed solutions.

There is no separate Part 2; schedule timed past-paper questions once you finish each block.

Part 1 (57 problems)

Advanced (14 problems)

This tier contains 14 problems aimed at extension and synthesis. Representative work includes "The Hyperbolic Tangent and Resisted Motion" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.

The Hyperbolic Tangent and Resisted Motion
The Non-Linear Oscillator and Separatrix Curves
Damped Harmonic Motion
Fourth-Order Verification
The Vibrating Cantilever Beam
Pharmacokinetics and Two-Compartment Diffusion
Radiosonde Telemetry in the Snowy Mountains
Non-Linear Differential Equations and Trigonometric Families
Non-Dimensionalisation and the Richards Growth Model
Verification of a Third-Order Solution
The Fourth-Order Oscillator (Enrichment)
Integrating Factor with Oscillatory Forcing
Isoclines and Optimisation on the Unit Circle
Logistic Growth and the Uniqueness Barrier

Basic (19 problems)

This tier contains 19 problems aimed at foundational fluency. Representative work includes "Verifying Solutions by Substitution" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.

Verifying Solutions by Substitution
Solving Initial Value Problems by Direct Integration
Implicit Differentiation and Conic Sections
Repeated Integration and Polynomial Families
Analyzing Solution Curves via Implicit Differentiation
Non-linear Slope Fields and Isoclines
The Geometry of Isoclines and Invariant Solutions
Trigonometric Curves and Auxiliary Angles
Separable Equations and Exponentials
Identifying the Vector Flow
The Periodic Flow
Conic Sections from Differential Equations
Direction Field Features and Hyperbolic Solutions
Initial Value Problem with a Square-Root Branch
Separable Equation and Horizontal Asymptote
Domain and Range from an Exponential DE
Reconstructing a Polynomial from a Direction Field
Implicit Curves and Domain of an IVP
Abstract Related Rates and Sign Analysis

Medium (24 problems)

This tier contains 24 problems aimed at exam-standard reasoning. Representative work includes "Transformations and Orthogonal Trajectories" — expect multi-step algebra, clear notation, and justification aligned with Mathematics Extension 1 and Extension 2 marking expectations.

Transformations and Orthogonal Trajectories
Separable Equations, Conics, and Domains
Singular Points and Self-Intersecting Curves
Homogeneous Differential Equations and Domain Boundaries
Inverse Trigonometric Identities and Restricted Domains
Picard Iterations and Series Convergence
Higher-Order Differential Equations and Real Roots
Advanced Initial Value Problems
Non-Linear Autonomous Initial Value Problems
Innovation Diffusion and Logistic Models
Newton's Law of Cooling in the Blue Mountains
Torricelli's Law and the Hurstville Plaza Basin
The Wollemi Bushfire Spread
The Rooftop Solar Boom in Western Sydney
Modelling Fish Population in a Protected Lake
Logistic Growth and Maximum Population Increase
Autonomous DE with Equilibria and Concavity
A Separable IVP with Logarithmic Domain Issues
Linear DE, Isoclines, and Local Minima
Orthogonal Trajectories of an Elliptic Family
A Tank-Filling Model with Evaporation
Qualitative Analysis of an Allee-Type Population Model
Inverse-Trigonometric IVP and Validity Interval
Exponential Blow-Up and a Vertical Asymptote

Common patterns across the booklet

Integration: 3 problems — e.g. "Verifying Solutions by Substitution"
Vectors & geometry: 1 problem — e.g. "Identifying the Vector Flow"
Mechanics: 2 problems — e.g. "The Hyperbolic Tangent and Resisted Motion"
Inequalities: 1 problem — e.g. "Homogeneous Differential Equations and Domain Boundaries"
Polynomials: 4 problems — e.g. "Integrating Factor with Oscillatory Forcing"
Trigonometry: 7 problems — e.g. "Non-Linear Differential Equations and Trigonometric Families"

Standout and less-seen problem types

These go beyond routine drills — expect unfamiliar wording or multi-topic synthesis:

The Hyperbolic Tangent and Resisted Motion: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
The Non-Linear Oscillator and Separatrix Curves: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
Damped Harmonic Motion: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
Fourth-Order Verification: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
The Vibrating Cantilever Beam: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.
Pharmacokinetics and Two-Compartment Diffusion: A multi-step question that combines syllabus ideas — worth attempting under timed conditions after you finish the fundamentals review.

Working through a large booklet

Review fundamentals before Part 1
Try each problem without hints, then compare solutions
Study appendices A–C for exam essentials; D–H for enrichment
Prerequisites: derivatives, integrals, basic physics context

Timed practice: allow about one minute per mark; write legible structure even when practising alone. Error log: tag mistakes as concept, algebra, or misreading. Rotation: do not camp on advanced tier if basics still slip.

Open HSC Differential Equations and work steadily — 57 problems is a marathon, not a sprint.

Key appendices

Quick reference — Formula sheet for common DE forms and methods. Use for quick reference or enrichment beyond routine exam questions.

Slope fields — How to read and sketch direction fields. Use for quick reference or enrichment beyond routine exam questions.

Modelling workflow — From word problem to equation, solution, interpretation. Use for quick reference or enrichment beyond routine exam questions.

Vector fields (enrichment) — Trajectories beyond the core syllabus. Use for quick reference or enrichment beyond routine exam questions.

Partial differentiation (enrichment) — PDE preview material. Use for quick reference or enrichment beyond routine exam questions.

Power series methods (enrichment) — Series solutions preview. Use for quick reference or enrichment beyond routine exam questions.

Characteristic equation (enrichment) — Linear DE preview. Use for quick reference or enrichment beyond routine exam questions.

Complex harmony (enrichment) — Links between DEs and complex numbers. Use for quick reference or enrichment beyond routine exam questions.

For HSC preparation, prioritise the first one or two appendices; later entries reward curious students but are not required for standard papers.

Key conclusion

Connect rate, graph, formula, and interpretation. Recent exams emphasise slope fields, separable equations, and logistic growth — solve carefully and explain meaning in words.

The booklet's closing section reinforces these habits:

Review fundamentals before Part 1
Try each problem without hints, then compare solutions
Study appendices A–C for exam essentials; D–H for enrichment
Prerequisites: derivatives, integrals, basic physics context

Revisit HSC Differential Equations in the fortnight before trials and redo problems you missed on first pass.

How to study with this booklet

Week 1 fundamentals; weeks 2–3 Part 1 warm-up and medium; week 4 advanced plus Appendix B slope fields; week 5 Mechanics booklet for motion links

General principles:

Closed-book first: attempt without notes, then check fundamentals.
Error log: record concept vs algebra vs reading errors.
Spaced repetition: redo missed questions after 3 and 7 days.
Past papers last: fix weak topics here, then sit full papers timed.

Mini-FAQ

Who is the HSC Differential Equations booklet for?

Extension 1 and 2 students, motivated learners, and teachers wanting concise reference plus enrichment studying Mathematics Extension 1 and Extension 2 under the NSW HSC.

Should I read solutions before attempting problems?

Attempt Part 1 first. Use Part 2 hints only after a genuine try or partial working.

Where can I read the booklet online?

Open HSC Differential Equations on Vu's Maths Hub — free, no account required.

How many problems are in the booklet?

Roughly 57 practice problems across 117 pages, each with worked solutions.

Is this aligned with NESA?

Topics match Mathematics Extension 1 and Extension 2 outcomes for exponential growth and decay, resistive motion, simple harmonic motion, slope fields, separable equations, and the logistic model. Confirm scope with your teacher and current NESA documentation.

Common mistakes to avoid

Separating variables without checking denominators for division by zero
Forgetting the constant of integration or initial condition
Misreading slope fields — sign of $\frac{d y}{d x}$ vs direction of solution curves
Giving only a formula without interpreting units or limiting behaviour
Rushing to advanced tiers before basic fluency — build foundations first.

Practice on Vu's Maths Hub

Open the free HSC Differential Equations on Vu's Maths Hub — 57 problems with full worked solutions.

Related resources:

How to use Vu's Maths Hub — Study DE before Mechanics
HSC Mechanics — Motion models using DEs

Exponential growth and decay — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, exponential growth and decay routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Differential Equations booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Differential Equations and locate items that stress exponential growth and decay; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Resistive motion and terminal velocity — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, resistive motion and terminal velocity routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Differential Equations booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Differential Equations and locate items that stress resistive motion and terminal velocity; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Escape velocity and gravitation — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, escape velocity and gravitation routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Differential Equations booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Differential Equations and locate items that stress escape velocity and gravitation; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Simple harmonic motion — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, simple harmonic motion routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Differential Equations booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Differential Equations and locate items that stress simple harmonic motion; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Water tanks and Torricelli models — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, water tanks and torricelli models routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Differential Equations booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Differential Equations and locate items that stress water tanks and torricelli models; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Slope fields — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, slope fields routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Differential Equations booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Differential Equations and locate items that stress slope fields; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Separable equations — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, separable equations routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Differential Equations booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Differential Equations and locate items that stress separable equations; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Logistic model — exam context

In NSW Mathematics Extension 1 and Extension 2 examinations, logistic model routinely appears as multi-mark questions where markers award method marks for clear setup. The HSC Differential Equations booklet builds this skill through dozens of graded problems — not one or two textbook examples. Open HSC Differential Equations and locate items that stress logistic model; attempt three without reading solutions first. When checking, compare structure (given/find, formula, substitution, answer in context) rather than only the final value. Log whether errors were misread questions, missing prerequisites, or algebra slips — that tag decides what to revise next.

Syllabus alignment

This booklet supports Mathematics Extension 1 and Extension 2 under the NESA syllabus. It supplements school instruction with 117 pages of extra exam-style practice — not a replacement for class teaching.

Additional exam advice

When sitting Mathematics Extension 1 and Extension 2 exams, allocate time proportional to marks. Practise concise justification in HSC Differential Equations — NSW markers reward clear communication. Reread the booklet conclusion the night before for a habit checklist.

Wednesday, 1 July 2026

HSC Differential Equations: Modelling, Slope Fields, and Separable ODEs