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

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: \( \frac{dy}{dx} = f(x) g(y) \).
• Rearrange: \( \frac{1}{g(y)} dy = f(x) dx \).
• Integrate both sides and include constant of integration.

Example -- Logistic style

Solve \( \frac{dy}{dx} = y(3 - y) \).

1. Separate: \( \frac{1}{y(3 - y)} dy = dx \).
2. Partial fractions: \( \frac{1}{y(3 - y)} = \frac{1}{3}\left( \frac{1}{y} + \frac{1}{3 - y} \right) \).
3. Integrate: \( \frac{1}{3}\bigl( \ln\lvert y \rvert - \ln\lvert 3 - y \rvert \bigr) = x + C \).
4. Solve for \( y \: \) \( \frac{y}{3 - y} = Ae^{3x} \) => \( y = \frac{3Ae^{3x}}{1 + Ae^{3x}} \).

Integrating Factor Method

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

Example -- Linear DE

Solve \( \frac{dy}{dx} + 2y = e^{-x} \).

1. \( P(x) = 2 \) => integrating factor \( \mu = e^{2x} \).
2. Multiply: \( e^{2x} \frac{dy}{dx} + 2 e^{2x} y = e^{x} \).
3. Recognise \( \frac{d}{dx}(e^{2x} y) = e^{x} \).
4. Integrate: \( e^{2x} y = \int e^{x} dx = e^{x} + C \).
5. 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 \( y'' + 4y' + 13y = 0 \).

1. Auxiliary: \( m^2 + 4m + 13 = 0 \) => \( m = -2 \pm 3i \).
2. Solution: \( y = e^{-2x}(A \cos 3x + B \sin 3x) \).

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 \( y'' - y = e^{x} \).

1. Complementary solution from \( m^2 - 1 = 0 \): \( y_c = Ae^{x} + Be^{-x} \).
2. Since \( e^{x} \) appears in \( y_c \), guess particular \( y_p = C x e^{x} \).
3. Substitute: \( y_p'' - y_p = C x e^{x} + 2C e^{x} - C x e^{x} = 2C e^{x} \).
4. Match RHS => \( 2C e^{x} = e^{x} \Rightarrow C = \tfrac{1}{2} \).
5. General solution: \( y = Ae^{x} + Be^{-x} + \tfrac{1}{2} x e^{x} \).

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: \( \frac{dT}{dt} = -k(T - T_\text{env}) \).

1. Solve separable equation to obtain \( T(t) = T_\text{env} + (T_0 - T_\text{env}) e^{-kt} \).
2. 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.

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: Review the probability modules in Section 6 to connect calculus techniques with statistics.