Differential equations connect a quantity to the way it changes, making them a powerful tool for modeling motion, growth, cooling, mixing, circuits, and many other real systems. Instead of only asking for a value, a differential equation describes a rule for change over time or space. This matters because many scientific laws are naturally written as rate statements before they are written as explicit formulas.
Learning to translate words into equations lets you build models rather than just use them.
Key Facts
- A differential equation relates an unknown function to one or more of its derivatives.
- A rate statement like the population grows at a rate proportional to its size becomes dP/dt = kP.
- If dy/dt = ky, then the general solution is y = Ce^(kt).
- If dy/dt = -k(y - A), then the solution is y = A + Ce^(-kt).
- An initial condition such as y(0) = y0 is used to find the constant C.
- The sign and units of a parameter help interpret the model, such as k in 1/s for a time-based rate.
Vocabulary
- Differential equation
- An equation that includes an unknown function and at least one derivative of that function.
- Rate of change
- A measure of how quickly a quantity changes with respect to another variable, often written as a derivative.
- Proportionality constant
- A constant that connects two proportional quantities, such as k in dP/dt = kP.
- Initial condition
- A known value of the function at a particular input, used to choose one solution from a family of solutions.
- Equilibrium solution
- A constant solution where the rate of change is zero, so the modeled quantity does not change.
Common Mistakes to Avoid
- Writing the amount instead of the rate is wrong because a phrase like grows at a rate proportional to P means dP/dt = kP, not P = kt.
- Ignoring the sign of change is wrong because decay, cooling, and draining usually require a negative rate when the quantity is above its target or reference level.
- Forgetting the initial condition is wrong because the differential equation gives a family of possible functions, and the initial value selects the specific model.
- Treating k as unitless is wrong because the proportionality constant must have units that make both sides of the differential equation match.
Practice Questions
- 1 A bacteria population grows at a rate proportional to its size. If P(0) = 500 and k = 0.30 per hour, write the differential equation and find P(t).
- 2 A cup of coffee cools in a room at 22 degrees Celsius according to dT/dt = -0.12(T - 22). If T(0) = 90, find T(t) and estimate T(10).
- 3 A model says dQ/dt = -0.4Q for the amount of medicine in the bloodstream. Explain what the negative sign means, what the equilibrium solution is, and how the graph should behave over time.