riaology

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Big idea

A differential equation doesn’t tell you what a quantity is. It tells you how it changes.
Your job is to turn that “rule for change” into a story about behavior over time.

START HERE

What is a differential equation, really?

In physics, chemistry, biology, and economics, the most useful facts often sound like: “when the system is bigger/hotter/farther from equilibrium, it changes faster.” Differential equations are the mathematical tools we use to describe those scenarios.

A DE is a rule for change

It’s a statement about slopes, not a single number.

A typical differential equation says:

“The rate of change of y depends on (t, y).”

In symbols:

\frac{dy}{dt}=f(t,y)

Quick check

choose one

Which sentence best matches what a DE gives you?

EXPLORE

Start like a researcher: what drives the change?

Explore: “what should the rate depend on?”

Choose a scenario. Before you “solve,” act like a researcher: decide what variables should control the change.

Scenario A — Growth

You track a bacteria culture. The bigger it gets, the faster it grows.

Prediction question

What should the growth rate depend on: time $t$ , or the current amount $y$ ?

Plausible “first models”

Real modeling is iterative. Start with a simple claim, then refine.

“Rate depends on how much you have” (proportional to y)

$\frac{dy}{dt}=ky,\ k>0$

“Rate depends only on the clock” (depends on t)

$\frac{dy}{dt}=g(t)$

FEEL IT

Slope first, solving later

Interactive: “a DE is a slope rule”

Let’s use $\frac{dy}{dt}=y$ . At each point $(t,y)$ , the slope equals the y-value. Drag the sliders and watch what that implies.

Point

$t=0,\ y=1$

What the DE says right here

Slope $\frac{dy}{dt}$ = 1 → the curve tilts up.

If we took a tiny step $dt=0.5$ , we’d predict:

y_{\text{next}} \approx y + \left(\frac{dy}{dt}\right)dt = 1 + (1)(0.5) = 1.5

WHY INFINITELY MANY?

Same law, different starting points

Key idea: one DE → many curves (until you add a starting value)

A differential equation is a rule for slopes. Many different curves can obey the same slope rule because you can start at different initial values.

Choose an initial condition

For $\frac{dy}{dt}=y$ , pick $y(0)=1$

Snapshot values along the curve

y(t)

slope

-2

0.14

-1.5

0.22

-1

0.37

-0.5

0.61

0.5

1.65

2.72

1.5

4.48

7.39

Notice: for $\frac{dy}{dt}=y$ , the slope equals the y-value. So the bigger y gets, the steeper the curve gets.

Key consequence

One DE usually allows infinitely many solution curves.
To pick one curve, you add an initial condition (like

y(0)=1

PRACTICE

Practice: predict first

Practice progress

1) Concept

keyword check

What does $\frac{dy}{dt}=f(t,y)$ claim about y and time?

Matched concepts: 0/4

Quick check

choose one

For population growth, what is usually the more reasonable dependency?

3) Uniqueness

keyword check

Why does a DE by itself usually not pick one solution curve?

Matched concepts: 0/3

Quick check

choose one

\frac{dy}{dt}

is positive whenever

y>0

, what does that suggest about solution curves above the x-axis?

Checkpoint

If you can explain “a DE is a rule for slopes, and an initial condition picks the curve,” you’re exactly where you need to be.

← Back to roadmap Next: Slope Fields →

What is a Differential Equation?

What is a differential equation, really?

Start like a researcher: what drives the change?

Slope first, solving later

Same law, different starting points

Practice: predict first