Level 3 Integration Walkthrough

2023 NCEA Level 3 Integration Question 3(e)

2023 Paper

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Question

Consider the differential equation

\[ (1-x^2)(1+y)\frac{dy}{dx}+(1-x)(1-y^2)=0. \]

Given that \(y=0\) when \(x=2\), find the value of \(y\) when \(x=6\).

First walkthrough idea

Focus to try first

Factor both \(1-x^2\) and \(1-y^2\), cancel the common factors carefully, then use the initial condition to choose the correct branch.

Step 1

Simplify the differential equation

Factoring \(1-x^2\) and \(1-y^2\) lets the common factors cancel cleanly.

Show the first step’s working

Factoring \(1-x^2\) and \(1-y^2\) lets the common factors cancel cleanly.

Key result

\[ \frac{dy}{dx}=-\frac{1-y}{1+x} \]

Walkthrough overview

What this question practises

This 2023 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Factor cancellation in a separable differential equation.

This is Question 3(e) from the 2023 NCEA Level 3 Integration paper for AS91579 — Apply integration methods in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91579.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise factor cancellation in a separable differential equation. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

← Question 3(d)