Introduction
This O-Level A-Maths question tests your understanding of exponential functions and differentiation.
You are required to differentiate a given function and then substitute values to form simultaneous equations in order to find unknown constants.
We follow the teacher’s exact working shown on the board, step by step.
The Question
Given that:
\(y = A e^{2x} + B e^{-x}\)
and that:
\(\dfrac{dy}{dx} + 4y = e^{2x} – e^{-x}\)
Find the values of constants \(A\) and \(B\).
Step-by-Step Working (Teacher’s Method)
Step 1: Differentiate \(y\)
\(\dfrac{dy}{dx} = 2A e^{2x} – B e^{-x}\)
Step 2: Substitute into the given equation
\((2A e^{2x} – B e^{-x}) + 4(A e^{2x} + B e^{-x}) = e^{2x} – e^{-x}\)
Step 3: Simplify
\(2A e^{2x} – B e^{-x} + 4A e^{2x} + 4B e^{-x} = 6A e^{2x} + 3B e^{-x}\)
So we get:
\(6A e^{2x} + 3B e^{-x} = e^{2x} – e^{-x}\)
Step 4: Compare coefficients
For \(e^{2x}\):
\(6A = 1 \;\;\Rightarrow\;\; A = \tfrac{1}{6}\)
For \(e^{-x}\):
\(3B = -1 \;\;\Rightarrow\;\; B = -\tfrac{1}{3}\)
✅ Final Answer
\(A = \tfrac{1}{6}, \quad B = -\tfrac{1}{3}\)
Key Concepts
| Concept | Why It Matters |
|---|---|
| Differentiation of \(e^{ax}\) | Brings down the coefficient \(a\) |
| Substitution | Links differentiation to algebra |
| Comparing coefficients | Fast and accurate way to find constants |
| Careful sign handling | Prevents common mistakes |
Tips for Students
- Always differentiate before substituting into equations
- Group terms with the same exponential base
- Compare coefficients carefully — signs matter
- Write each step clearly to avoid algebra errors
For Parents
Questions like this develop strong links between calculus and algebra — essential skills for O-Level A-Maths success.
Mastering these techniques builds confidence for more advanced calculus topics.