We have the rule (dp/dt = αp). Now we want the solution — an actual formula for the population at any time t. To solve a differential equation analytically, we integrate both sides.

First, the initial condition. A differential equation tells you how things change, but not where they started. So we must pin down a starting value. Here, the population at time 0:

p(0) = P₀

Different starting values P₀ give different solutions — but they don't change the underlying equation. That reusability is a key strength of differential equations.

Now solve it. We want a function whose derivative equals α times itself. Rearrange and integrate both sides:

∫ (1/p) dp = ∫ α dt → ln(p) = α·t + C → p(t) = P₀ · e^(α·t)

Integrating 1/p gives the natural log; then we exponentiate both sides to free p. Steps explained just below.

Wait — where did that ln suddenly come from? You don't calculate it, you recognize it. Integration is just differentiation run backwards, so ∫ (1/p) dp is really asking: "what function has 1/p as its derivative?" From the last tutorial, the derivative of ln(p) is 1/p — so its integral must be ln(p). You're reading a derivative you already know, in reverse.

Why is 1/p the weird one that needs a log? The power rule says ∫ pⁿ dp = pⁿ⁺¹ / (n+1). Try it on 1/p (which is p⁻¹, so n = −1): you'd get p⁰ / 0 — divide by zero, it breaks! There's a hole in the power rule exactly at n = −1, and the function that plugs that hole is the natural log. That's the one place ln shows up — remember it as: the 1/x exception → ln.

The last step: undo the log by raising e to _both sides_. e and ln are opposites — they cancel, like × and ÷ (e^(ln p) = p). So starting from ln(p) = α·t + C:

• raise e to both sides → p = e^(α·t + C)
• split the exponent → p = e^C · e^(α·t)
• rename the leftover constant e^C as the starting population P₀

…and you land on the clean solution:

p(t) = P₀ · e^(α·t)

'Population grows (or shrinks) exponentially as a function of time and birth rate.'

📓 The only integrals worth memorizing (they're just the differentiation rules from last tutorial, read backwards):
• ∫ xⁿ dx = xⁿ⁺¹/(n+1) — the power rule
• ∫ (1/x) dx = ln|x| — the special exception (n = −1)
• ∫ eˣ dx = eˣ — itself!
• ∫ sin x dx = −cos x and ∫ cos x dx = sin x

Unlike derivatives (always mechanical — product/chain rule), integrals have no guaranteed recipe: you pattern-match these few, then for anything harder you look it up (Wikipedia's "Lists of integrals") or let SymPy do it exactly — sp.integrate(1/p, p) returns log(p).

Why an _exponential_? This is the beautiful payoff from the last tutorial. Remember eigenfunctions — the exponential is the one function whose derivative equals itself (times a constant). The equation demanded a function proportional to its own rate of change, and the exponential is the only thing that fits. The eigenfunction lesson wasn't a detour — it was setup for this exact moment.

Reality check: most differential equations do not have a clean solution like this. The population equation is a lucky, friendly case. When there's no exact formula, we go numerical (next quest). A small historical aside the source loves: a lot of these integration "tricks" were invented by mathematicians writing each other taunting letters bragging they could solve something the others couldn't. So don't stress about memorizing tricks.

Enough algebra — now watch it move. Below are the same two graphs from this section, but live and side by side: the dp/dt line (the equation) on the left and the p(t) curve (the solution) on the right. One slider controls everything: α, the birth rate. Drag it and watch both respond together.

Drag α and you'll find three regimes — this is the real reason to play with it:
1. α > 0 → the line slopes up, the solution explodes exponentially toward infinity (runaway growth).
2. α < 0 → the line slopes down, the solution decays to zero — a stable end state.
3. α = 0 → the line is flat at zero (dp/dt = 0). Nothing changes, ever. This special spot is the equilibrium point (also called a stable point or fixed point).

Lock in 'equilibrium point.' It's any state where the rate of change is zero, so the system sits still. Picture a ball at the bottom of a bowl — no push, no motion. We're about to meet the exact same idea in a neuron, where the equilibrium is its resting voltage.

Where the model breaks (and secretly wins): real populations can't grow forever — food and space run out, so growth eventually levels off; and this model ignores weather, predators, disease. But its cousin, the Drift-Diffusion Model, uses nearly the same equation to model how your brain accumulates evidence to make a decision. Same math, wildly different use — that's the power of differential equations.