7.1 Modeling Situations with Differential Equations

What is a differential equation?

A differential equation is an equation involving a function and its derivative(s). Instead of solving for a number, you solve for a function. In calculus, these often describe how a quantity changes over time, like population growth, cooling temperatures, or radioactive decay.

Rate of change language

Word problems translate into equations using phrases like 'rate of change is proportional to' → dy/dx = ky. 'Proportional to the difference between' signals Newton's Law of Cooling: dT/dt = k(T − T_room). Learning to translate English into symbols is the key modeling skill.

Verifying solutions

To check if a function is a solution to a differential equation, substitute the function and its derivative(s) into the equation and confirm both sides are equal. This works even without solving the equation — just plug in and simplify using derivative rules.

Key terms

Differential equation

An equation relating a function to one or more of its derivatives.

Solution to a differential equation

A function that, when substituted (with its derivative) into the equation, makes both sides equal.

dy/dx = ky

Models a quantity whose rate of change is proportional to the quantity itself (exponential growth/decay).

Newton's Law of Cooling

dT/dt = k(T − T_room): rate of temperature change is proportional to the difference between object and surrounding temperature.

Translating word problems

Key phrases like 'proportional to' or 'rate of change' signal how to set up the differential equation from a real-world description.

Try an AP-style question

A population grows at a rate proportional to its size. Which equation models this?

1. dP/dt = kP
2. dP/dt = k
3. dP/dt = kt
4. P = kt