FURTHER MATHEMATICS · DIFFERENTIAL EQUATIONS

Classify the equation.
Build the right solution family.

A structured guide for Singapore A-Level Further Mathematics syllabus 9649. Connect characteristic roots to solution form, then use conditions without losing constants.

Study the worked example Try a differential-equation question

Equation classification
Auxiliary roots
Initial-condition checks

01Constants select the curve Initial or boundary conditions choose one member of the solution family.

Built around 9649 Further MathematicsFirst- and second-order differential equations and applications

Structure drives formRoot type determines the complementary solution

THE CLASSIFICATION HABIT

Order, linearity, homogeneity

Before solving, identify the highest derivative, whether y and its derivatives appear linearly, and whether the right-hand side is zero.

For a homogeneous linear equation with constant coefficients, exponential trial solutions turn differentiation into algebra.

ROOTS TO SOLUTIONS

Let the auxiliary equation choose the form

Distinct real roots

y = A e^{m_{1} x} + B e^{m_{2} x}

Repeated real root

y = (A + B x) e^{m x}

Complex roots

y = e^{α x} (A \cos β x + B \sin β x)

Non-homogeneous?

Add a suitable particular integral to the complementary function.

y = y_{c} + y_{p}

WORKED EXAMPLE

Find the roots, then determine the constants.

Solve $\frac{d^{2} y}{d x^{2}} - 3 \frac{d y}{d x} + 2 y = 0$ , given y(0)=1 and y′(0)=0.

EXAM HABIT

Write the general solution before applying initial conditions.

1
Form the auxiliary equation
$m^{2} - 3 m + 2 = 0$
2
Find the roots and solution
$m = 1, 2 \Rightarrow y = A e^{x} + B e^{2 x}$
3
Apply both conditions
$A + B = 1, A + 2 B = 0$
Hence A=2 and B=−1.
4
State the particular solution
$y = 2 e^{x} - e^{2 x}$

Answery = 2eˣ − e²ˣ

COMMON MISTAKES

What to catch before the examiner does

Wrong repeated-root form

A repeated root needs the independent term x eᵐˣ.

Applying conditions too soon

Build the complete general solution before solving for constants.

Losing coefficients when differentiating

Differentiating e²ˣ produces 2e²ˣ.

Skipping verification

Substitute the solution and check every supplied condition.

PRACTISE WITH FEEDBACK

Make every solution-form choice explicit.

Integrand can inspect your root classification, constants and verification.

Try Integrand

1
Classify
Identify equation and roots.
2
Construct
Write the general solution.
3
Verify
Check equation and conditions.

QUICK QUESTIONS

Further Maths differential equations FAQ

Why use an auxiliary equation?

For constant-coefficient homogeneous linear equations, an exponential trial turns each derivative into multiplication by a power of m.

When do I use an x multiplier?

Use it to create a linearly independent term when an auxiliary root is repeated, and when a particular-integral trial overlaps the complementary function.

Is this guide within the Further Maths syllabus?

Yes. First- and second-order differential equations are included in Singapore A-Level Further Mathematics syllabus 9649.

READY TO PRACTISE?

Classify, solve and verify with feedback.

Open Integrand

Classify the equation.Build the right solution family.

Order, linearity, homogeneity

Let the auxiliary equation choose the form

Distinct real roots

Repeated real root

Complex roots

Non-homogeneous?

Find the roots, then determine the constants.

Form the auxiliary equation

Find the roots and solution

Apply both conditions

State the particular solution

What to catch before the examiner does

Wrong repeated-root form

Applying conditions too soon

Losing coefficients when differentiating

Skipping verification

Make every solution-form choice explicit.

Classify

Construct

Verify

Further Maths differential equations FAQ

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Classify, solve and verify with feedback.

Classify the equation.
Build the right solution family.