Level 3 Integration Walkthrough

2025 NCEA Level 3 Integration Question 3(c)

2025 Paper

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Question

Consider the differential equation \[ \frac{dy}{dx}=\frac{\sqrt{4y+1}}{x^2}. \]

Given that \(y=2\) when \(x=\frac{2}{3}\), find the value(s) of \(y\) when \(x=\frac{4}{5}\).

You must use calculus and show the results of any integration needed to solve the problem.

First walkthrough idea

Hint to try first

Move the radical involving \(y\) to the left and the power of \(x\) to the right.

Step 1

Separate the variables

This puts all the \(y\)-terms on one side and all the \(x\)-terms on the other.

Show the first step’s working

This puts all the \(y\)-terms on one side and all the \(x\)-terms on the other.

Key result

\[ (4y+1)^{-1/2}\,dy=x^{-2}\,dx \]

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91579 — Apply integration methods in solving problems.

Method: Solving a separable differential equation with a radical.

This is Question 3(c) from the 2025 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 solving a separable differential equation with a radical. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

State or check the real-domain restriction and test for extraneous solutions after squaring.

Continue practising

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