Differential equations are where H2 calculus meets the real world — in H2 Mathematics every examinable equation is the separable first-order type, solved by separating the variables, integrating both sides, then fixing the arbitrary constant from a given condition. Students who keep those steps in order find the topic far more predictable than it first appears. This guide is from Ancourage Academy, whose JC H2 Mathematics tuition teaches differential equations step by step in small groups of 3–6 at Bishan and Woodlands.
This is a single-topic deep-dive — a sibling to our H2 Math calculus and sequences and series guides, and part of our wider H2 Mathematics overview. Differential equations build directly on integration, so secure that first.
If differential equations are a gap, Ancourage Academy's JC2 H2 Mathematics programme drills the separable-equation method directly — book a trial class (usually $18) for a diagnostic assessment.
What Do Differential Equations Cover in H2 Math?
In H2 Mathematics (9758), differential equations covers a single examinable form — the first-order separable equation dy/dx = f(x)g(y) — solved by separating the variables and integrating both sides, reducing an equation to that form using a given substitution, and formulating and interpreting differential equations that model rate-of-change situations. The SEAB Mathematics syllabus (9758) defines what is examinable; second-order equations and the integrating-factor method belong to other syllabuses, not 9758, and any substitution needed is always provided in the question.
What Forms of Differential Equation Are Examinable?
H2 Mathematics tests one form — the first-order separable equation, written dy/dx = f(x)g(y), where the right-hand side factorises into a function of x times a function of y.
- Separable, dy/dx = f(x)g(y): separate the variables and integrate both sides (the method below).
- The special case dy/dx = f(x): this is just the separable form with g(y) = 1, solved by integrating once, y = ∫f(x) dx + C.
- Reducible by a given substitution: a question may supply a substitution that turns a non-separable-looking equation into the separable form.
Each integration introduces one arbitrary constant, so a first-order equation needs exactly one given condition to pin it down. Second-order equations such as d²y/dx² = f(x), and the integrating-factor method, are not part of the 9758 syllabus — you never integrate twice or build an integrating factor. Forgetting the arbitrary constant is the most common single error in the whole topic.
How Do You Solve a Separable Equation?
A first-order equation is separable when you can write it so that all the y terms are on one side and all the x terms on the other — then you integrate both sides.
- Separate: rearrange dy/dx = g(x)h(y) into (1/h(y)) dy = g(x) dx.
- Integrate both sides — remember the single arbitrary constant.
- Make y the subject if the question asks for the explicit solution.
When a substitution is provided (for example, "use the substitution u = …"), differentiate the substitution, replace the terms in the original equation, solve the resulting separable equation in the new variable, then substitute back. The substitution is always given — you are never expected to invent it.
What Is the Difference Between General and Particular Solutions?
The general solution contains the arbitrary constant(s) and represents a whole family of solution curves; the particular solution uses given conditions to fix the constant(s) and pick out one specific curve.
| Term | Meaning |
|---|---|
| General solution | Includes the arbitrary constant(s) — a family of curves |
| Particular solution | Constant(s) found from given conditions — one curve |
| Family of solution curves | Sketch showing the general solution for several constant values |
When asked to sketch a family of solution curves, show how the curves shift as the constant changes and include any equilibrium or asymptotic behaviour — this is a graph-sketching question as much as a calculus one, drawing on functions and graphs skills.
How Are Differential Equations Used in Modelling?
Real-world rate-of-change problems are translated into a differential equation, solved, and then interpreted — the most common being exponential growth and decay, where the rate is proportional to the amount present.
- Exponential model: "the rate of change is proportional to the quantity" becomes dx/dt = kx, whose solution is exponential.
- Limited growth / cooling: "the rate is proportional to the difference from a fixed value" leads to models such as Newton's law of cooling.
Translating the words into the correct equation is half the question — read carefully for "proportional to," the sign of k (growth vs decay), and the meaning of any constants in context.
The Most Common Differential Equations Mistakes
In our H2 Math classes at Ancourage Academy, a handful of recurring errors cause most avoidable mark loss in this topic.
| Mistake | Why it happens | How to fix it |
|---|---|---|
| Omitting the arbitrary constant | Treating it like a definite integral | Add the constant C when you integrate, before applying the given condition |
| Not separating fully | Leaving an x or y on the wrong side | Confirm only y-terms with dy and only x-terms with dx before integrating |
| Wrong sign of k | Misreading growth as decay | Decay means the quantity falls, so dx/dt is negative — check the context |
| Forgetting to substitute back | Leaving the answer in the substituted variable | After solving in u, replace u with the original expression |
| Mishandling the constant after a logarithm | Leaving ln |y| without exponentiating correctly | From ln |y| = g(x) + C, write y = A e^(g(x)), absorbing the constant into A |
How Do Differential Equations Connect to the Rest of H2 Math?
Differential equations are applied calculus, and they pull in graphing and earlier algebra.
- Calculus: integration is the engine of every solution. See our calculus deep-dive.
- Functions and graphs: sketching families of solution curves uses graph-sketching skills. See our functions and graphs guide.
- Partial fractions: some separable equations need a fraction decomposed before integrating. See our A-Maths partial-fractions guide.
A Study Plan for Mastering H2 Differential Equations
Work this topic in order: integration revision, then separable equations, then given substitutions, then modelling.
- Week 1 — integration revision: make sure integration techniques are fluent, since every solution relies on them.
- Week 2 — separable equations: drill separating the variables and integrating both sides, tracking the arbitrary constant.
- Week 3 — given substitutions: practise reducing equations to separable form using the substitution provided in the question.
- Week 4 — modelling: translate worded rate-of-change problems into equations and interpret the solutions.
Ancourage Academy's JC1 and JC2 H2 Mathematics programmes work through differential equations on exactly this progression in small groups of 3–6. Book a trial class (usually $18) for a diagnostic, or WhatsApp us with any questions.
Common Questions About H2 Math Differential Equations
What types of differential equation does H2 Math test?
H2 Mathematics tests one form: the first-order separable equation dy/dx = f(x)g(y), solved by separating the variables and integrating both sides. The special case dy/dx = f(x) is simply g(y) = 1, integrated once. A question may provide a substitution that reduces an equation to the separable form. It also tests forming and interpreting differential equations from real-world rate-of-change situations such as exponential growth and decay. Second-order equations and integrating factors are not in the 9758 syllabus.
What is the difference between a general and a particular solution?
The general solution contains the arbitrary constant(s) from integration and represents an entire family of solution curves. The particular solution uses given conditions — such as a known value of y at a particular x — to find the constant(s) and identify one specific curve. A first-order separable equation has a single arbitrary constant, so one given condition is enough to fix it and pick out one specific solution curve.
How do you solve a separable differential equation?
Rearrange dy/dx = g(x)h(y) so that all the y terms are with dy on one side and all the x terms are with dx on the other: (1/h(y)) dy = g(x) dx. Integrate both sides, adding a single arbitrary constant, then make y the subject if an explicit solution is required. If the equation is not directly separable, apply the substitution given in the question, solve, and substitute back.
How do you model exponential growth with a differential equation?
When a quantity changes at a rate proportional to its current amount, the relationship is dx/dt = kx, where k is a constant. Separating and integrating gives an exponential solution. A positive k models growth and a negative k models decay, so read the context carefully to set the sign. Translating the worded statement into the correct equation is where most marks are won or lost.
Related: H2 Mathematics Overview · H2 Math Calculus · Functions & Graphs · Sequences & Series · A-Maths Partial Fractions