Key Idea: A differential equation links a quantity to its rate of change, like a rumour spreading. Separation of variables solves it, and a starting value fixes the constant.
Separation of variables
- Recognise — dy/dx = f(x)·g(y).
- Separate — ∫ (1/g(y)) dy = ∫ f(x) dx.
- Integrate — Both sides, with one + C.
- Find C — From the initial condition, straight away.
- Finish — Make y the subject, interpret with units, and check the long-term behaviour.
| Step | Result |
|---|---|
| dN/dt = kN | N = A eᵏᵗ |
| N = 4 at t = 0 | A = 4 |
| N = 16 at t = 2 | k = ln 2 ≈ 0.693 |
| t = 6 | N = 4·2⁶ = 256 |
Put every y with dy and every x with dx, then integrate.
Is dy/dx = x + y separable? No. It is a sum, not an x-part times a y-part.
Why is 256 students after 6 hours only a rough guess? Pure exponential growth never stops. Near the size of the whole school, a logistic model fits better.
dy/dx = 2xy. Separate it. ∫ (1/y) dy = ∫ 2x dx.
How many constants do you add when you integrate both sides? Just one.