ZIMSEC A Level · Pure Mathematics · Form 6
A differential equation links a function with its derivatives.
First-order separable equations
Rearrange to g(y) dy = f(x) dx, then integrate both sides. Use given conditions to find the constant.
Worked example: Solve dy/dx = 2xy, given y = 3 when x = 0.
∫(1/y) dy = ∫2x dx → ln y = x² + c
x = 0, y = 3: c = ln 3 → y = 3e^(x²)
Forming equations from words
“The rate of decrease of a quantity N is proportional to N” gives dN/dt = −kN, so N = N₀e^(−kt).
Worked example: The temperature θ of a cooling liquid satisfies dθ/dt = −k(θ − 20). Initially θ = 90 °C, and after 10 minutes θ = 55 °C. Find θ after 20 minutes.
θ − 20 = 70e^(−kt). At t = 10: 35 = 70e^(−10k) → e^(−10k) = ½
At t = 20: θ − 20 = 70 × (½)² = 17.5 → θ = 37.5 °C
Interpreting solutions
Describe long-term behaviour: in the cooling example, θ → 20 °C as t → ∞ (room temperature).
Check yourself
- Solve dy/dx = y/x given y = 2 when x = 1.
- Write a differential equation: the rate of growth of a population P is proportional to P.
- Solve dy/dx = e^(−y) given y = 0 when x = 0.
Show answers
- ln y = ln x + c → y = 2x
- dP/dt = kP (k > 0)
- eʸ dy = dx → eʸ = x + 1 → y = ln(x + 1)
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