Differential equations are a relatively small part of the Specialist Mathematics course, but they carry disproportionate weight in exams. The 2023 and 2024 Examination 1 and Examination 2 papers show very clearly that VCAA uses differential equations to assess depth of understanding rather than procedural fluency. Students who treat these questions as integration exercises often lose marks even when their working appears correct.
What VCAA is actually assessing with differential equations
According to the Study Design, differential equations are assessed for students’ ability to model relationships, apply conditions, and interpret solutions in context. This intent is visible across both years. Questions are rarely limited to “solve the differential equation”. Instead, students are asked to form a differential equation from a description, apply initial or boundary conditions, and explain what the resulting function represents.
The Examiner’s Reports repeatedly note that students struggle most not with solving the equation, but with the steps before and after the solution.
Forming the differential equation correctly
In both the 2023 and 2024 papers, several questions required students to translate a physical or contextual description into a differential equation. This was a major point of discrimination. Many students attempted to guess the form of the equation rather than deriving it logically from the information given.
The Examiner’s Reports highlight that students often wrote down a differential equation that looked plausible but did not reflect the relationship described in the question. For example, confusing proportionality with inverse proportionality, or omitting constants that were explicitly required. These errors were conceptual, and marks were lost early in the solution as a result.
High-scoring responses typically included a brief statement explaining why the differential equation had a particular form before solving it. This made the reasoning explicit and easier to reward.
Applying initial conditions accurately
Another consistent source of error was the application of initial conditions. Many students correctly solved a general differential equation but then applied the condition incorrectly, substituted the wrong value, or failed to solve fully for the constant.
In both years, Examiner’s Reports note that students sometimes treated initial conditions as optional, leaving constants unevaluated or partially evaluated. In Specialist Mathematics, this is treated as an incomplete solution. The final expression must satisfy the given condition exactly.
Students who wrote out the substitution step carefully and solved explicitly for the constant were far more likely to access full marks.
Interpretation is where most marks are lost
The most revealing part of the Examiner’s Reports concerns interpretation. In several 2023 and 2024 questions, students were required to explain what a solution meant in context, such as describing how a quantity changed over time or identifying long-term behaviour.
Many students stopped once they had an equation. Marks were lost because they did not explain what the solution implied. For example, failing to comment on growth or decay, equilibrium behaviour, or limiting values. These explanations were not add-ons. They were central to what the question was assessing.
The reports consistently show that students who interpreted solutions clearly, using appropriate mathematical language, were rewarded even when minor algebraic slips were present elsewhere.
CAS does not remove the reasoning requirement
In Examination 2, CAS was often used to assist with integration or solving. However, the Examiner’s Reports emphasise that CAS output alone was insufficient. Students who relied on CAS without showing how the solution was obtained, or without applying conditions and interpreting the result, lost marks.
High-scoring students used CAS as a tool to support their reasoning, not as a substitute for it. They still articulated each step and explained how the CAS result fitted into the solution.
Why differential equations feel deceptively difficult
Differential equations feel manageable in practice because they are often taught as structured procedures. In exams, that structure is removed. Students must decide what the equation should be, how to solve it, and how to interpret it, all within a single question. This is why the topic is so effective at separating levels of understanding.
The Study Design expects students to reason with differential equations, not just solve them. The exams reflect this expectation very consistently.
How students can improve their performance
Improvement comes from practising explanation, not more integration. Students should practise writing out why a differential equation has a particular form, how conditions are applied, and what solutions mean in context. Reviewing past questions alongside the Examiner’s Reports is especially valuable here, because the reports identify exactly where reasoning broke down.
An ATAR STAR perspective
At ATAR STAR, differential equations are taught as a modelling language rather than a technique. We help students slow down, justify each step, and communicate meaning clearly. This approach supports students who feel that differential equations should be straightforward but keep costing them marks, as well as high-performing students refining precision and interpretation.
In Specialist Mathematics, differential equations reward clarity of thought more than speed of execution. That is exactly how VCAA intends them to function.