Partial Fractions Technique: Solving Differential Equations Clearly

Partial Fractions Technique – What Examiners Look For

✏️ Differential equations that involve partial fractions are rarely difficult because of the calculus itself. The difficulty comes from sequencing. Students know how to separate variables. They know how to integrate. They know how to use partial fractions. What often goes wrong is when each of those skills is used.

This is why these questions appear so frequently in exams. They are a clean way of testing whether students can apply developing A Level Maths techniques in a controlled order rather than as isolated tricks. Examiners are not looking for speed or flair. They are looking for structure that holds together from the first line to the last.

This blog focuses on differential equations exam technique, specifically how partial fractions should be used when solving separable equations, and why small timing errors cost far more marks than students expect.

🔙 Previous topic:

Before working through differential equations with partial fractions, it helps to revisit finding the common ratio in complex geometric sequences, because both topics reward careful algebra and logical sequencing rather than quick shortcuts.

🧭 Why partial fractions appear in differential equations

When students meet partial fractions earlier in the course, they are usually treated as a self-contained algebra skill. In differential equations, that separation disappears. Partial fractions are no longer “the point of the question”. They are simply a tool that allows integration to happen.

Consider an equation like
$\displaystyle \frac{dy}{dx}=\frac{2x+1}{x(x+1)}.$

At first glance, this looks ready to integrate. Many students instinctively reach for logarithms. Examiners expect stronger candidates to pause here. The expression in
$\displaystyle x$
is not integrable yet. Something has to be done before calculus begins.

Recognising that pause is part of the exam skill.

📘 What examiners expect you to recognise early

There are two decisions examiners are looking for in the opening lines. The first is whether the equation is separable. The second is whether the separated expression is integrable as it stands.

If all the
$\displaystyle y$
terms can be placed on one side and all the
$\displaystyle x$
terms on the other, separation is correct. After that, the examiner’s attention shifts immediately to algebra. If the integrand is a rational function, partial fractions are usually unavoidable.

Students who try to integrate too early often produce answers that look sophisticated but are structurally wrong. Examiners are trained to spot this quickly.

🧮 Worked Exam Example (Done Correctly)

📄 Exam Question

Solve the differential equation
$\displaystyle \frac{dy}{dx}=\frac{2x+1}{x(x+1)}$
given that
$\displaystyle y=0 \text{ when } x=1.$

✏️ Full Solution (Exam-Style)

Start by separating the variables:
$\displaystyle dy=\frac{2x+1}{x(x+1)},dx.$

At this point, integration is not possible. Rewrite the right-hand side using partial fractions:
$\displaystyle \frac{2x+1}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}.$

So,
$\displaystyle 2x+1=A(x+1)+Bx.$

Comparing coefficients gives:
$\displaystyle A=1,\quad B=1.$

Rewrite the equation:
$\displaystyle dy=\left(\frac{1}{x}+\frac{1}{x+1}\right)dx.$

Integrate both sides:
$\displaystyle \int dy=\int\left(\frac{1}{x}+\frac{1}{x+1}\right)dx.$

This gives:
$\displaystyle y=\ln|x|+\ln|x+1|+C.$

Now apply the condition
$\displaystyle y=0 \text{ when } x=1.$

$\displaystyle 0=\ln1+\ln2+C \Rightarrow C=-\ln2.$

Final solution:
$\displaystyle y=\ln|x|+\ln|x+1|-\ln2.$

⚠️ The same question — how it often goes wrong

A very common student attempt looks like this:

They separate correctly, then immediately try to integrate:
$\displaystyle \int \frac{2x+1}{x(x+1)},dx.$

Some students then split the fraction informally or guess at logarithms without setting up partial fractions. Others apply the initial condition before finding a general solution, creating unnecessary algebra early on.

From an examiner’s point of view, this is where marks start to leak. Once the structure breaks, it becomes difficult to award method marks even if the final answer resembles the correct one.

This is why examiners emphasise order. Not because they are picky, but because correct order makes reasoning visible.

📌 Method Mark Breakdown

When I mark this question, the first thing I’m checking is whether the student has recognised separation correctly. Writing
$\displaystyle dy=\frac{2x+1}{x(x+1)}dx$
earns an immediate method mark. It tells me they know what kind of problem this is.

The next thing I’m looking for is restraint. A student who sets up partial fractions before integrating shows awareness that calculus is not yet possible. That earns another method mark even before any numbers are calculated.

Integration is where marks are often protected. Writing both integrals clearly and including a constant
$\displaystyle C$
is essential. Forgetting the constant usually prevents full marks later, even if everything else is correct.

Finally, accuracy marks come from applying the given condition at the correct stage and producing a clean final expression. Students often “do the hard maths” correctly and then lose marks by applying the condition too early or carelessly.

This is a classic example of why structure matters more than cleverness.

🧠 Why this fits into bespoke A Level Maths revision advice

Questions like this reward students who understand sequencing, not just techniques. Revision that treats partial fractions, integration, and differential equations as separate topics misses the point. Exams test how they link.

Students who practise whole solutions, including the pauses between steps, build reliability much faster than those who practise fragments in isolation.

🎯 If geometric sequences keep costing you marks

Dependability comes from order, not confidence. These questions only feel risky when steps are taken out of sequence.

Examiners expect a fixed structure:

always separate variables first
always rewrite the integrand fully before integrating
always complete the integration before applying initial conditions

That order should never change, even when the algebra looks tempting. Skipping or rearranging steps is what turns a standard method-mark question into a fragile one.

The reason this works so well is that it protects marks. Even if coefficients are wrong, a clear and consistent structure allows examiners to award follow-through marks. Disorder removes that safety net.

With repeated practice, this sequence stops feeling like a checklist and becomes automatic. This is exactly what an online A Level Maths Revision Course with full examples is designed to reinforce — not just how to start a question, but how to carry it cleanly to the end every time.

✅ Conclusion

Differential equations involving partial fractions reward calm, staged working because every mark scheme is built around a chain of decisions. The examiner is not just checking whether you can integrate. They are checking whether you can recognise what needs to happen before integration becomes possible. That is why separation comes first, and why rewriting the integrand sits in the middle of the solution rather than at the end.

When you separate cleanly, you make the method obvious. The moment you try to integrate too early, you usually create algebra that is harder to control and harder to mark. Partial fractions are not there to make the question “long”. They are there to make the integral achievable in a way that is transparent. The best scripts look almost boring because the steps are predictable: rewrite, integrate, then tidy.

Applying conditions is where reliability is either confirmed or lost. Initial conditions are not an extra line you add at the end for formality. They are the step that shows you understand what the constant represents. The constant is not decoration. It is the whole family of solutions you created by integrating. The condition chooses the one curve the question is describing, and examiners want to see that choice made at the right stage, not guessed early.

In other words, examiners are not looking for shortcuts. They are looking for solutions that stay logically connected from the first line to the last. If your working is easy to follow, method marks are easy to award. If your working becomes compressed, jumpy, or inconsistent, marks disappear even when your final answer looks “close”. The aim in these questions is not to look clever. It is to be reliably markable under pressure.

✍️ Author Bio

👨‍🏫 S. Mahandru

An experienced A Level Maths teacher with extensive UK exam-board familiarity, specialising in calculus, algebraic structure, and examiner-focused exam technique.

🧭 Next topic:

When rearranging and separating variables in differential equations, you must differentiate expressions cleanly and confidently — so if that algebra ever feels uncertain, revisit A Level Differentiation – Complete Exam Guide (All Techniques) to strengthen the foundations that make the method work.

❓ FAQs

🧠 Why do partial fractions make differential equations feel so unreliable in exams?

They feel unreliable because they interrupt the flow students expect. Many students approach differential equations thinking, “separate and integrate.” Partial fractions force a pause, and that pause often triggers panic.

The presence of extra constants such as
$\displaystyle A$
and
$\displaystyle B$
can also feel distracting under pressure. Students start worrying about algebra instead of structure. Examiners see this every year.

Once students accept that partial fractions are simply a preparation step, the anxiety drops. The equation is not harder — it is just staged. Understanding that staging is the key to confidence.

⚠️ Do examiners allow follow-through if partial fractions are wrong?

In many cases, yes — but this is where students often misunderstand what “follow-through” really means. Examiners are not checking whether your partial fractions are numerically perfect in isolation. They are checking whether your thinking stays consistent from one line to the next. If your decomposition is set up correctly and you then integrate your own expression logically, examiners can usually follow that chain of reasoning.

For example, if you correctly recognise that partial fractions are needed and write
$\displaystyle \frac{2x+1}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1},$
but then make a small slip when finding the values of
$\displaystyle A$
and
$\displaystyle B,$
that does not automatically destroy the solution. If you integrate the resulting terms correctly and carry that same expression through consistently, method marks are often still available.

Where follow-through stops is when the working becomes unclear or inconsistent. If values change without explanation, if terms appear or disappear between lines, or if the student abandons their own decomposition partway through, the examiner has nothing solid to reward. At that point, they cannot tell whether later steps come from understanding or from guessing.

This is why examiners repeatedly stress “clear working” in reports. They are not asking for neatness for its own sake. They are asking for continuity. A solution that is slightly wrong but logically connected is far more markable than a solution that jumps between ideas.

The safest mindset is this: once you commit to a decomposition, you must commit to it fully. Even if you suspect it might be wrong, stay consistent. Examiners can follow a wrong path, but they cannot follow a disappearing one.

🎯 How do I make these questions dependable rather than risky?