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H2 Maths Notes 5 5 Differential Equations

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H2 Maths Notes (JC 1-2): 5.5) Differential Equations

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07 Oct 2025, 00:00 Z

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Q: What does H2 Maths Notes (JC 1-2): 5.5) Differential Equations cover?
A: First-order, second-order, and modelling techniques for H2 Maths Topic 5.5.

Before you revise
Recognise the differential-equation family before you start: separable, integrating factor, or second-order linear. Write a short checklist for each method and verify solutions by differentiating them.

First-Order Separable Equations

Form: $\dfrac{\mathrm{d}y}{\mathrm{d}x} = f(x) g(y)$ .
Rearrange: $\dfrac{1}{g(y)} {\mathrm{d}}y = f(x) {\mathrm{d}}x$

Part 22 of 32

Previous topicH2 Maths Notes (JC 1-2): 5.4) Definite Integrals Next topicH2 Maths Notes (JC 1-2): 5) Calculus

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Integrate both sides and include constant of integration.

Example -- Logistic style

Solve $\dfrac{\mathrm{d}y}{\mathrm{d}x} = y(3 - y)$ .

Separate: $\dfrac{1}{y(3 - y)} {\mathrm{d}}y = {\mathrm{d}}x$ .
Partial fractions: $\dfrac{1}{y(3 - y)} = \dfrac{1}{3}\left( \dfrac{1}{y} + \dfrac{1}{3 - y} \right)$ .
Integrate: $\dfrac{1}{3}\bigl( \ln\lvert y \rvert - \ln\lvert 3 - y \rvert \bigr) = x + C$
Solve for $y \:$ $\dfrac{y}{3 - y} = Ae^{3x} \implies y = \dfrac{3Ae^{3x}}{1 + Ae^{3x}}$

Integrating Factor Method

Linear form: $\dfrac{\mathrm{d}y}{\mathrm{d}x} + P(x) y = Q(x)$ .
Integrating factor $\mu(x) = e^{\int P(x) {\mathrm{d}}x}$ .
Multiply entire equation by $\mu(x)$ , recognise left side as derivative of $\mu(x) y$ .

Example -- Linear DE

Solve $\dfrac{\mathrm{d}y}{\mathrm{d}x} + 2y = e^{-x}$ .

$P(x) = 2 \implies$ integrating factor $P(x) = 2 \implies$ .
Multiply: $e^{2x} \dfrac{\mathrm{d}y}{\mathrm{d}x} + 2 e^{2x} y = e^{x}$ .
Recognise $\dfrac{d}{\mathrm{d}x}(e^{2x} y) = e^{x}$ .
Integrate: $e^{2x} y = \int e^{x} {\mathrm{d}}x = e^{x} + C$
Solution: $y = e^{-x} + C e^{-2x}$ .

Second-Order Linear Equations

Homogeneous: $y'' + ay' + by = 0$ .
Auxiliary equation $m^2 + am + b = 0$ .
Distinct real roots $m_1, m_2$ : solution $y = A e^{m_1 x} + B e^{m_2 x}$
Repeated root $m$ : $y = (A + Bx) e^{mx}$ .
Complex roots $\alpha \pm i\beta$ : $y = e^{\alpha x}(A \cos \beta x + B \sin \beta x)$

Example -- Damped oscillator

Solve $y'' + 4y' + 13y = 0$ .

Auxiliary: $m^2 + 4m + 13 = 0$ => $m = -2 \pm 3i$ .
Solution: $m = -2 \pm 3i$ .

Particular Integrals

For non-homogeneous $y'' + ay' + by = R(x)$ , guess form based on $R(x)$ .
Use annihilator or undetermined coefficients; adjust guess if it overlaps with complementary function.

Example -- Forcing term

Solve $y'' - y = e^{x}$ .

Complementary solution from $m^2 - 1 = 0$ : $y_c = Ae^{x} + Be^{-x}$ .
Since $e^{x}$ appears in $y_c$ , guess particular $y_p = C x e^{x}$
Substitute: $y_p'' - y_p = C x e^{x} + 2C e^{x} - C x e^{x} = 2C e^{x}$
Match RHS $\implies 2C e^{x} = e^{x} \implies C = \dfrac{1}{2}$
General solution: $y = Ae^{x} + Be^{-x} + \dfrac{1}{2} x e^{x}$

Modelling with Differential Equations

Translate physical laws (cooling, population growth, circuits) into DEs.
Apply initial conditions to determine constants.
Check units and interpret solutions physically (long-term behaviour, equilibrium).

Example -- Cooling

Newton cooling: $\dfrac{dT}{dt} = -k(T - T_\text{env})$ .

Solve separable equation to obtain $T(t) = T_\text{env} + (T_0 - T_\text{env}) e^{-kt}$ .
Use given temperature at time $t$ to determine $k$ .

Verifying Solutions

Differentiate proposed solution and substitute back into original equation.
For initial value problems, check both DE and initial condition(s).
Discuss domains where solution is valid (avoid division by zero, etc.).

Calculator Workflow

GC diffEQ or numerical solver checks solutions; always show analytical steps first.
Use GC to plot solution behaviour with chosen constants for modelling tasks.
For exponential solutions, store constants to evaluate quickly at required times.

Exam Watch Points

State integrating factor explicitly and show multiplication step.
Include constants of integration and apply boundary/initial conditions.
Present final solutions neatly, indicating complementary and particular parts when relevant.
Comment on long-term behaviour when the question frames a modelling context.

Practice Quiz

Ensure you can separate variables, apply integrating factors, and interpret modelling constants confidently.

Quick Revision Checklist

Separate variables and integrate confidently for first-order DEs.
Apply integrating factor method and solve resulting linear equations.
Handle second-order homogeneous and non-homogeneous equations using auxiliary equations and particular integrals.
Interpret solutions within modelling contexts, including initial conditions and long-term trends.

Next steps: Keep the H2 Maths notes hub open and move into Section 6 - Probability & statistics to link calculus with inference work.

H2 Maths Notes (JC 1-2): 5.5) Differential Equations

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