First-order Differential Equations
Solving linear first-order equations with integrating factors and forming general or particular solutions for growth, decay and coupled models.
How to study A-level Further Mathematics
Define the objects and conditions, select a representation, carry out exact mathematics, then validate, interpret and communicate the result.
Core concepts
Concept 1
A first-order linear equation is placed in standard form before constructing its integrating factor.
Exam cue: Divide through to make the coefficient of the derivative one.
Concept 2
The general solution contains an arbitrary constant; a boundary or initial condition determines a particular solution.
Exam cue: Recognise the transformed left side as the derivative of the integrating factor times the dependent variable.
Concept 3
Coupled first-order systems can model interacting quantities such as predator and prey populations.
Exam cue: Interpret signs, equilibria and long-term behaviour in the original context.
Risk pitfalls and guardrails
Using the wrong sign in the integrating factor.
Guardrail: Do not replace proof with examples, exact reasoning with unverified calculator output, or a valid awarding-body route with an invented mix of options.
Applying an initial condition before the general solution is complete.
Guardrail: Do not replace proof with examples, exact reasoning with unverified calculator output, or a valid awarding-body route with an invented mix of options.
Solving algebraically without checking the model's permitted values.
Guardrail: Do not replace proof with examples, exact reasoning with unverified calculator output, or a valid awarding-body route with an invented mix of options.
Memory anchors
First-order Equation
A first-order differential equation contains a first derivative as its highest derivative.
Integrating Factor
An integrating factor converts a linear first-order equation into a product derivative.
General Solution
A general solution contains the arbitrary constant representing a family of curves.
Particular Solution
A particular solution satisfies stated boundary or initial conditions.
Equilibrium
An equilibrium is a constant state at which the model's rates of change vanish.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Solve dy/dx+2y=eˣ using an integrating factor. Which factor is required?
Find the general solution of dy/dx+2y=eˣ.
Answer all questions to submit.
Next step personalized recommendations
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Move forward only after this module is stable.
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