Free lesson · Differential equations

Start with a rate rule and an initial value

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Start with the idea

A differential equation gives a rate of change.
An initial value supplies the starting level.

Symbols, units & horizon

y(t): amount at time t in units
t: elapsed hours
c: fixed rate in units/hour
y₀: starting amount in units
dy/dt: rate of change of y with time

When and why to use this

Use rate equations to model inventory, balances or physical systems over time. Proportional growth and mean reversion build on this example.

Read $\frac{d y}{d t} = 2$ as “y increases by 2 units per hour.”
Use $y (0) = 5$ to specify the starting value.
After one hour: 7. After two hours: 9.
A solution gives the value at every allowed time.
Check a solution by differentiating it and checking its starting value.

\frac{d y}{d t} = c, y (0) = y_{0} ⟹ y (t) = y_{0} + c t

Core rule · definition and worked arithmetic

Start with a rate rule and an initial value

Constant accumulation over time t is $c t$ .
Add the initial amount: $y (t) = y_{0} + c t$ .
Differentiate: the constant term contributes 0 and ct contributes c.
At t=0 the formula returns y₀, satisfying the initial condition.

Work it by hand

With c=2 and y₀=5, y(3)=5+2×3=11. With the same rate but y₀=8, the answer becomes 14.

Use the rule

Name the inputs and units.
Work the small example by hand.
Check the result before continuing to the next lesson.

Before moving on

Explain the core rule in one sentence, reproduce the worked calculation and solve both practice variations.

Research sources, review dates and limitations

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def constant_rate_path(initial, rate, elapsed):
    return initial+rate*elapsed

print(constant_rate_path(5,2,3))

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