Differential Equations NCEA Practice Questions | Calc.nz

Skill overview

What these questions practise

The matched Integration methods explicitly name separation or a differential equation, while matched Differentiation methods verify or prove a differential-equation relationship.

Relevant standards: NCEA Level 3 Differentiation — AS91578 · NCEA Level 3 Integration — AS91579

This collection contains 23 matching walkthroughs across 9 examination years.

The grouping is based on the method recorded for each walkthrough. Use the linked NZQA standard and paper material for authoritative wording, diagrams, and assessment information.

Before you start

Common mistakes in these methods

Separate the variables consistently before integrating, without dropping factors that belong with either differential.
Retain the integration constant and apply the supplied condition, then check the resulting branch and domain in context.
For verification, calculate the required derivatives and substitute them into both sides of the stated differential equation.

Related practice

Differential Equations questions from different years

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NCEA Level 3 · AS91578 2024 Differentiation 1 question

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Question 2(c) — Higher Derivatives First and second derivatives in a proof-style differential-equation question.

NCEA Level 3 · AS91578 2023 Differentiation 1 question

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Question 3(e) — Differential Equations Verifying a differential equation from a catenary model.

NCEA Level 3 · AS91578 2020 Differentiation 1 question

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Question 3(e) — Differential Equations Proving a differential equation using first and second derivatives.

NCEA Level 3 · AS91578 2017 Differentiation 1 question

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Question 3(e) — Higher Derivatives Second derivatives and a differential-equation parameter.

NCEA Level 3 · AS91579 2025 Integration 2 questions

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Question 1(e) — Differential Equations Separating variables and evaluating the resulting model. Question 3(c) — Differential Equations Solving a separable differential equation with a radical.

NCEA Level 3 · AS91579 2024 Integration 3 questions

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Question 1(d) — Differential Equations Product-to-sum before integrating a differential equation. Question 2(c) — Differential Equations Separating variables with an exponential model. Question 3(b) — Differential Equations Solving a logarithmic differential equation using a condition.

NCEA Level 3 · AS91579 2023 Integration 3 questions

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Question 1(d) — Differential Equations Separating variables and using a condition to keep the correct solution branch. Question 2(b) — Differential Equations Integrating a radical differential equation and fitting the constant. Question 3(e) — Differential Equations Factor cancellation in a separable differential equation.

NCEA Level 3 · AS91579 2022 Integration 1 question

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Question 2(c) — Differential Equations Separating variables and using a condition in a log model.

NCEA Level 3 · AS91579 2021 Integration 2 questions

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Question 2(e) — Differential Equations Solving a separable differential equation to find a vertical distance. Question 3(c) — Differential Equations Separating a trigonometric differential equation and using a condition.

NCEA Level 3 · AS91579 2020 Integration 2 questions

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Question 1(e) — Differential Equations Separating a logarithmic trigonometric differential equation. Question 2(d) — Differential Equations A separable radical and trigonometric differential equation.

NCEA Level 3 · AS91579 2019 Integration 2 questions

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Question 1(e) — Differential Equations Proving an exponential-growth relation from a differential equation. Question 3(b) — Differential Equations Integrating a sec squared (2x) differential equation.

NCEA Level 3 · AS91579 2018 Integration 2 questions

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Question 1(b) — Differential Equations Integrating a differential equation and using an initial condition. Question 1(d) — Differential Equations Separating variables and applying an initial condition.

NCEA Level 3 · AS91579 2017 Integration 2 questions

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Question 3(d) — Differential Equations Solving a separable differential equation with a condition. Question 3(e) — Integration by Substitution Substitution in a parameterised differential equation.

Keep practising

Antidifferentiation Integration techniques

Page updated 9 August 2026.