Here's the whole idea of Euler's method in one move: wherever a differential equation says dy/dt, just replace it with the slope-of-a-line and rearrange. That turns an unsolvable equation into a simple recipe for stepping forward.

Take the population equation dp/dt = a·p. Swap the derivative for the two-point slope:

(p₁ − p₀) / Δt = a·p₀ → p₁ = p₀ + Δt·(a·p₀)

p₀ = the population you know now; p₁ = your estimate a step Δt later.

📌 Memorize this sentence — the whole chapter rides on it:

Future = current + (time-step × dynamics)

You take where you are right now, add a little nudge in the direction the equation points, and that's your guess for a moment later. Then repeat. Forever.

It's exactly how a ship navigates in fog with no landmarks (dead reckoning): I know where I am and how fast I'm going, so I can estimate where I'll be in one minute — then re-measure and step again.

🧭 Fun fact: this is named after Leonhard Euler (1707–1783), one of the most prolific mathematicians ever — he kept publishing even after going blind. His little stepping trick is still the backbone of nearly every physics engine, weather model, and neuron simulation today.