Ask Yourself First
Before we write a single equation, let's think physically. Imagine a colony of bacteria in a petri dish, right now, at this moment.
How fast the population grows depends on how large the population already is. More bacteria → more splitting → faster growth. The rate of change is proportional to what's already there.
That one sentence — "the rate of change is proportional to the current amount" — is the entire physical content of this section. Everything else is just making it precise.
The same story plays out in radioactive decay, compound interest, cooling objects, and drug clearance. One idea, infinite applications.
Imagine every tiny chunk of $y$ has a chance to reproduce or disappear. If $k = 0.1$, each unit grows by 10% per unit time. If $k = -0.1$, each unit shrinks by 10% per unit time. So $k$ is literally the per-unit rate of change — a rate that every piece of the quantity experiences simultaneously and identically.
This is why the equation shows up in so many places: atoms decay independently, money compounds independently, signal power attenuates per unit length. Exponential behavior = each piece acting independently and identically.
The ODE From Scratch
Let $y(t)$ be the quantity we're tracking — population, charge, mass, dollars, whatever. We want to describe how it changes over time.
Step 1 — State the Core Assumption
We assume the rate of change is proportional to how much is already present. In math notation, proportionality looks like this:
The $\propto$ symbol means "is proportional to." This is the entire physical claim. It's not pulled from thin air — it's a modeling assumption that matches how bacteria split, how atoms decay, how interest accrues.
Step 2 — Turn Proportionality Into an Equation
A proportionality always becomes an equation by inserting a constant of proportionality. If $A \propto B$, then $A = kB$ for some constant $k$. Apply that here:
That's it. That's the growth/decay ODE. Every symbol in it comes directly from the physical assumption — nothing is added arbitrarily. The constant $k$ is the only thing we had to introduce, and it has a clear job: it converts the proportionality into an equality by specifying exactly how much the current amount drives the rate of change.
Proportionality ($\propto$) tells you the shape of the relationship — that $dy/dt$ and $y$ move together. But it says nothing about the scale. Is each bacterium splitting once per hour or once per day? Is the decay fast or slow? The constant $k$ is what pins down that scale. It encodes the physics.
What Each Symbol Means
| Symbol | Physical meaning |
|---|---|
| $y(t)$ | The quantity at time $t$ — population, mass, dollars, charge… |
| $dy/dt$ | The instantaneous rate of change — how fast $y$ is growing or shrinking right now |
| $k$ | The growth constant — encodes how strongly the current amount drives the change. $k>0$: growth. $k<0$: decay. |
Think of $k$ as a slope amplifier. The ODE says: wherever you are on the $y$-axis, your slope at that moment equals $k$ times your height. If $k = 0.3$ and $y = 100$, you're rising at 30 units per unit time. Climb higher, slope gets steeper. It's a self-reinforcing feedback loop.
Think of $k$ as controlling the system's personality:
$k > 0$ → Growth. Bigger $y$ drives a bigger increase. Snowball effect. Examples: bacteria doubling, compound interest, chain reactions.
$k < 0$ → Decay. Bigger $y$ drives a faster decrease. Self-extinguishing. Examples: radioactive decay, capacitor discharge, drug clearance.
$k = 0$ → Frozen. No change. $y$ stays constant forever.
The magnitude $|k|$ tells you how aggressively the system evolves — a large $|k|$ means fast response, a small $|k|$ means sluggish change.
Visualizing the Slope Field
Every point $(t, y)$ has a slope arrow equal to $ky$. Notice: arrows only depend on $y$, not $t$. Horizontal slices have constant slope — this is an autonomous ODE. The chart below shows multiple solution curves overlaid on the field.
First Principles of k
Before we solve the ODE, let's interrogate the one symbol we haven't fully pinned down. You might be tempted to read $k$ as "just a number that scales $y$" — but that's underselling it. Let's build up exactly what $k$ is from scratch.
Step 1 — The Core Assumption (Again, But Deeper)
Start from the physical claim: the rate of change is proportional to the current amount.
The $\propto$ symbol only tells you the shape — that $dy/dt$ and $y$ move together. It says nothing about the scale. Is each bacterium splitting once per second or once per decade? That's unknown so far. To turn a proportionality into an equation, you introduce a constant that carries the scale:
So $k$ is born as the proportionality constant — the number that converts "amount present" into "rate of change." That's its entire job description.
Step 2 — Interrogate the Units
Check units — this is the physicist's move. $\dfrac{dy}{dt}$ has units of $\dfrac{[\text{stuff}]}{[\text{time}]}$. And $y$ has units of $[\text{stuff}]$. So:
Solving for $k$:
$k$ has units of inverse time — like $\text{hr}^{-1}$, $\text{s}^{-1}$, $\text{yr}^{-1}$. It is literally a rate per unit time, per unit of $y$. Not a scaling factor. Not an amplitude. A fractional rate.
$k = 0.1\,\text{hr}^{-1}$ means: each unit of $y$ contributes $0.1$ units of change per hour. If you have $y = 300$, your rate is $0.1 \times 300 = 30$ units/hr.
Step 3 — Concrete Numerical Intuition
Let's make this visceral with numbers. Say $y = 100$ and $k = 0.1$:
Now change $k$. The amount $y$ doesn't change — just how aggressively each piece of $y$ drives the rate:
| $k$ | $dy/dt$ at $y=100$ | Behavior |
|---|---|---|
| $k = 0.5$ | $50$ units/time | Fast growth 📈 |
| $k = 0.1$ | $10$ units/time | Slow growth 📈 |
| $k = 0$ | $0$ units/time | Frozen ─ |
| $k = -0.2$ | $-20$ units/time | Decay 📉 |
Think of $k$ not as "modifying $y$" but as a per-unit contribution rate:
"Each unit of $y$ contributes $k$ units of change per unit time."
So if you have twice as much $y$, you get twice as much rate. If $k$ is large, each piece is acting aggressively. This is exactly the proportionality assumption, rephrased as a physical mechanism.
Step 4 — The Sign of k as Physics
The sign of $k$ isn't a mathematical accident — it encodes the physical direction of feedback:
| Sign | What it means physically | Real example |
|---|---|---|
| $k > 0$ | More $y$ → faster increase → runaway growth | Bacteria, compound interest, chain reactions |
| $k < 0$ | More $y$ → faster decrease → self-extinguishing decay | Radioactive decay, RC discharge, drug clearance |
| $k = 0$ | Amount has no effect on rate — system is frozen | No physical system (degenerate case) |
For EE specifically: when you see $k = -1/RC$ in a capacitor discharge, the negative sign is saying "the more charge you have, the faster you lose it." The physics is in the sign.
Step 5 — Why Multiply by y? The Proportionality Argument
Someone might ask: why does $\dfrac{dy}{dt}$ depend on $y$ itself? Why not just $\dfrac{dy}{dt} = k$? Let's compare the two models:
| Model | What it says | Example |
|---|---|---|
| $\dfrac{dy}{dt} = k$ | Constant rate, regardless of how much $y$ there is | Filling a pool at a fixed tap flow |
| $\dfrac{dy}{dt} = ky$ | Rate scales with how much $y$ is present | Each bacterium splits → more bacteria → more splits |
The second model encodes a feedback loop: the more you have, the faster you gain (or lose) more. That feedback is what creates the exponential curve. Without it, you'd get a straight line. The multiplication by $y$ is the mathematical expression of "each piece acts independently and identically."
You might have originally thought: "$k$ is there to modify $y$." That's close — but the deeper truth is:
$k$ is the proportionality constant that converts "amount present" into "instantaneous rate of change."
Its units are $[\text{time}]^{-1}$. Its sign encodes direction of feedback. Its magnitude encodes the aggressiveness of the system. It is not a scaling amplitude — it is a fractional rate, experienced equally by every piece of $y$ simultaneously. That's why doubling $y$ doubles $dy/dt$: every piece contributes, all at once.
Separation of Variables
We want to find $y(t)$. The ODE $\dfrac{dy}{dt} = ky$ is separable — we can group all "$y$ stuff" on one side and all "$t$ stuff" on the other.
-
Separate the variables. Divide both sides by $y$, multiply by $dt$:
$$\frac{1}{y}\,dy = k\,dt$$We moved all $y$-dependence left, all $t$-dependence right. Each side talks about one variable only.
-
Integrate both sides.
$$\int \frac{1}{y}\,dy = \int k\,dt \implies \ln|y| = kt + C$$Left side: $\ln|y|$ because $\int \frac{1}{y}\,dy = \ln|y|$. Right side: $kt + C$.
-
Exponentiate both sides to undo the $\ln$:
$$|y| = e^{kt+C} = e^C \cdot e^{kt} \implies y = A\,e^{kt}$$$A = \pm e^C$ absorbs the sign and the constant.
-
Apply the initial condition. At $t=0$, $y = y_0$:
$$y_0 = A\,e^{0} = A \implies A = y_0$$
$y_0$ = initial amount at $t=0$. $k>0$: growth. $k<0$: decay.
Growth case ($k > 0$): starts slow, then accelerates, then explodes. This isn't a coincidence — because the rate is proportional to the size, a larger $y$ produces an even larger derivative, which makes $y$ grow even faster. The curve is always "chasing its own tail upward."
Decay case ($k < 0$): drops fast at first, then slows down, then asymptotically approaches zero. It never quite reaches zero — because as $y$ shrinks, the rate of decrease shrinks too. The curve is always "braking as it falls."
Look at the ODE again: $\dfrac{dy}{dt} = ky$. The right side depends only on $y(t)$ — the current value. Not on the history of $y$. Not on how long the system has been running. Not on where it came from.
This is what mathematicians call a Markov property and what physicists call memorylessness: the future is entirely determined by the present. There are no delays, no history, no inertia from the past. Just:
"What is $y$ right now?" → "That determines everything."
This is why exponential decay describes radioactive atoms (each atom decays with the same probability regardless of its age), why it describes RC discharge (the capacitor doesn't "remember" how long it's been charging), and why it describes signal attenuation (each meter of cable attenuates the same fraction).
Why is $e$ the "right" base? (Deep Dive)
We could write $y = y_0 \cdot 2^{t/\tau}$ or use any other base — but $e$ is special because it's the unique base where $\frac{d}{dt} e^t = e^t$. The ODE literally says "my derivative is proportional to me" — so $e^{kt}$ is the only function that satisfies it naturally.
Concretely: $k$ is the fractional growth rate. $k = 0.03$ means you gain 3% of whatever you currently have, continuously, per unit time.
Interactive Explorer
Use the sliders to build intuition. Notice: $k$ controls the shape of growth/decay, and $y_0$ just scales the whole curve vertically.
Solving for k
In real problems you'll often be given two data points and asked to find $k$ — working backwards from data to discover the underlying physics.
- Write the model with the known point plugged in.
$$y_1 = y_0\,e^{k\,t_1}$$
- Isolate the exponential. Divide both sides by $y_0$:
$$\frac{y_1}{y_0} = e^{k\,t_1}$$
- Take the natural log of both sides.
$$\ln\!\left(\frac{y_1}{y_0}\right) = k\,t_1$$
- Solve for $k$.
$$\boxed{k = \frac{1}{t_1}\ln\!\left(\frac{y_1}{y_0}\right)}$$
The ratio $y_1/y_0$ tells you how much the quantity multiplied. The $\ln$ converts that multiplicative factor into an additive rate. Dividing by $t_1$ spreads it over the time interval to get a per-unit-time rate.
Half-Life and Doubling Time
| Concept | Definition | Formula | Sign of $k$ |
|---|---|---|---|
| Doubling time $T_2$ | $y(T_2) = 2y_0$ | $T_2 = \dfrac{\ln 2}{k}$ | $k > 0$ |
| Half-life $T_{1/2}$ | $y(T_{1/2}) = \tfrac{1}{2}y_0$ | $T_{1/2} = \dfrac{\ln 2}{|k|}$ | $k < 0$ |
Derivation of the Half-Life Formula
$\ln 2 \approx 0.693$ appears universally in exponential decay — radioactive isotopes, drug half-lives, RC circuit time constants.
When It's Not $ky$
The ODE $\dfrac{dy}{dt} = ky$ is the simplest model where change is proportional to amount. It's valid when two conditions hold: (1) the rate depends only on current amount, and (2) there are no limits or interactions. When either breaks down, the equation needs modification.
Logistic Growth — When Space Runs Out
Bacteria can't grow forever in a petri dish — eventually they run out of food. The growth rate gets suppressed as $y$ approaches a carrying capacity $L$:
When $y \ll L$: the factor $(1 - y/L) \approx 1$, so we recover pure exponential growth. When $y \to L$: the factor $\to 0$, and growth halts. The curve looks like an S — fast growth in the middle, asymptotic at both ends. Pure $ky$ is the limiting case of this when $L \to \infty$.
RC Circuit — $k$ Tied to Physics
A capacitor discharging through a resistor obeys:
This is still $\dfrac{dy}{dt} = ky$ — just with $k = -1/RC$ determined by physical constants. The structure is identical; the physics fixes the value of $k$. This is the pattern you'll see throughout EE: the same ODE, but $k$ derived from component values instead of from biological or financial data.
Use $\dfrac{dy}{dt} = ky$ when: rate depends only on current amount and no caps, limits, or cross-interactions. When those fail, you need a modified equation — but the intuition of $ky$ is always the starting point you perturb from.
Real-World Problems
Example 1 — Bacterial Growth
A culture starts with 500 bacteria. After 3 hours there are 2000.
(a) Find $k$. (b) How many after 8 hours? (c) When does population reach 10,000?
Full Solution — Bacteria Problem
Part (a):
Part (b):
Part (c):
Example 2 — Radiocarbon Dating
Carbon-14 has a half-life of 5730 years.
(a) Find $k$. (b) A sample has 80% of its original $^{14}$C. How old is it?
Full Solution — Radiocarbon Dating
Part (a):
Part (b):
Interactive Solver
Enter any two known data points — get the full step-by-step solution and a plotted curve.
📐 Growth/Decay Calculator
Given $y(0)=y_0$ and $y(t_1)=y_1$, find $k$ and evaluate at $t^*$.
Why This Matters for EE
As an electrical engineer, you'll see $\frac{dy}{dt} = ky$ everywhere in disguise:
| System | Equation | Solution | $k$ value |
|---|---|---|---|
| RC discharge | $\dfrac{dV}{dt} = -\dfrac{V}{RC}$ | $V(t) = V_0\,e^{-t/RC}$ | $k = -1/RC$ |
| RL current rise | $\dfrac{di}{dt} = -\dfrac{R}{L}i$ | $i(t) \sim e^{-Rt/L}$ | $k = -R/L$ |
| Signal decay | $\dfrac{dA}{dt} = -\alpha A$ | $A(t) = A_0\,e^{-\alpha t}$ | $k = -\alpha$ |
The time constant $\tau = -1/k$ tells you how fast the system responds. After one $\tau$, you're at $e^{-1} \approx 36.8\%$ of the original. After $5\tau$, within 1% of zero. This is the language of RC circuits, filter cutoffs, and signal processing — all built on $e^{kt}$.
The Core Framework
- The physical idea: rate of change ∝ current amount → $\dfrac{dy}{dt} = ky$
- The solution: $y(t) = y_0\,e^{kt}$ (by separation of variables)
- Finding $k$ from data: $k = \dfrac{1}{t_1}\ln\!\left(\dfrac{y_1}{y_0}\right)$
- Half-life / doubling time: $T = \dfrac{\ln 2}{|k|}$
- To predict: plug $k$, $y_0$, and $t$ back into $y(t) = y_0\,e^{kt}$
Exponential behavior = "each piece acts independently and identically."
That's why it shows up everywhere. Atoms decay independently. Money compounds independently. Signals attenuate per unit length. RC capacitors discharge — each unit of charge driving the same fractional exit rate. Once you see this, you'll recognize $e^{kt}$ not as a formula you memorized, but as the only honest answer to the question: "what happens when each part of a system has a constant fractional rate of change?"
Next: derive half-life from scratch → connect to RC time constants → build toward Circuits I and Signals & Systems.