Differential Equations: What You Need to Know for Ext 1 vs Ext 2

 

Intro

Extension 1 covers basic first-order differential equations and growth/decay models; Extension 2 extends to modelling in Mechanics and more rigorous solution techniques. Know which course owns each method so you do not over-study or miss a syllabus dot-point. Aligned with NESA syllabus Extension 1 and 2. Keywords: HSC differential equations syllabus, Ext 1 vs Ext 2 math, solve DE problems.

Summary

Separable equations dy/dx = f(x)g(y) and exponential growth dN/dt = kN are Extension 1 staples. Extension 2 applies DEs to velocity, resisted motion, and force models in Mechanics. Always apply initial conditions and interpret solutions in context with correct units.

Separable equations appear in population models, cooling laws, and radioactive decay in Extension 1. Extension 2 mechanics turns the same technique toward velocity and displacement with physical interpretation required in every answer line.

Key Points

• Extension 1: separable DE, exponential growth/decay, interpreting solutions with initial conditions.
• Extension 2: DE as models for motion — links directly to the Mechanics topic.
• Setting up the DE from a word problem requires a clear sign convention and force diagram.
• Check solutions by substitution and note domain restrictions (e.g. t ≥ 0).
• Separable form: ∫dy/g(y) = ∫f(x) dx — show both integrations for method marks.
• Read HSC Differential Equations booklet before Mechanics.

Worked example

Question. A quantity N satisfies dN/dt = −0.2N, with N(0) = 500. Find N(t) and the time when N = 250.

Solution.

1. Separable: dN/N = −0.2 dt.
2. Integrate: ln|N| = −0.2t + C, so N = Ae^−0.2t.
3. Initial condition: 500 = A, hence N(t) = 500e^−0.2t.
4. Set N = 250: 250 = 500e^−0.2t, so e^−0.2t = 1/2.
5. −0.2t = ln(1/2) = −ln 2, giving t = 5 ln 2 ≈ 3.47 (3 s.f.).

Answer. N(t) = 500e^−0.2t; t = 5 ln 2.

Takeaway. Exponential decay questions reduce to finding A from initial data, then solving a simple exponential equation.

Exam Preparation

Revise separable DE and exponential models first — they appear in both courses. Extension 2 students should then connect DE setup to Mechanics word problems. Do at least four modelling questions where you derive the DE from scratch, not just solve a given equation.

When revising, create two columns on one page: Extension 1 techniques on the left, Extension 2 modelling extensions on the right. Separable equations appear in both columns; only the right column includes motion and force language. This visual map prevents studying Extension 2-only material when your immediate exam is Extension 1.

1. Map syllabus scope. List Extension 1 vs Extension 2 DE outcomes from NESA and tick your confidence.
2. Practise setup and solve. Alternate between given-DE questions and word-problem setup.
3. Link to Mechanics. After each DE session, attempt one velocity or resisted-motion problem.

Word problems dominate applied DE marks: identify the rate of change, define variables with units, and state the initial condition in the same sentence as the DE. Extension 1 students should practise interpreting parameters — what does k > 0 mean in decay? Extension 2 students link velocity DEs to displacement by integration when asked. Always verify by differentiating your solution and substituting back into the original equation before boxing the answer.

Mini-FAQ

Is integration by parts in DE scope?

Extension 1 focuses on separable and exponential forms. Extension 2 Mechanics may require integration techniques you learned in calculus — not new DE methods beyond modelling.

Do I need to include absolute value when integrating 1/N?

Show ln|N| initially; if N > 0 by context (population, mass), state that and drop the absolute value.

Can the same DE appear in Ext 1 and Ext 2?

The solving technique may overlap, but Extension 2 questions add physical interpretation and multi-step modelling.

Common mistakes to avoid

• Forgetting the constant of integration before applying initial conditions.
• Wrong sign in exponential decay (using +k instead of −k).
• Giving decimal answers when exact form (e.g. 5 ln 2) is clearer.
• Skipping units or context in applied DE questions.

Practice on Vu's Maths Hub

Need more practice on this topic? Open the free HSC Differential Equations booklet on Vu's Maths Hub — worked examples and exam-style questions, readable in your browser with no account required.

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