Modeling Situations with Differential Equations

Equations that describe change itself

• Many real laws describe not a quantity, but how fast it changes.
• A differential equation 微分方程 relates a function to its derivatives.
• "Population grows at a rate proportional to its size" becomes $\dfrac{dP}{dt}=kP$.
• Solving it recovers the function; this unit is about writing and solving such equations.

Translating words into a rate equation

• The phrase "the rate of change of $y$" is $\dfrac{dy}{dt}$ — the left side of your equation.
• "proportional to $y$" → $ky$; "proportional to the difference from $M$" → $k(M-y)$.
• Match the English description of the rate to an expression in $y$ (and maybe $t$).
• The whole model is: (rate of change) = (that expression).

A cooling cup of coffee: its rate of cooling depends on how much hotter it is than the room

Common modeling phrases

• Proportional to the amount: $\dfrac{dy}{dt}=ky$ — exponential growth/decay.
• Proportional to the amount present and remaining: $\dfrac{dy}{dt}=ky(M-y)$ — logistic-type.
• Constant rate: $\dfrac{dy}{dt}=c$.
• Reading the phrase tells you the right-hand side.

A differential equation has a family of solutions

• Unlike an algebraic equation (one unknown number), a differential equation has an unknown function.
• Its solution is usually a family of functions (with a constant), not a single answer.
• A specific member is pinned down later by an initial condition (lesson 7.7).
• For now, the goal is just to write the equation from the situation.

Write from a proportional rate

• The rate of change of y is proportional to y.
• The constant of proportionality is k.