Differential Equations - Equations That Describe Change
A differential equation is an equation that involves a function and one or more of its derivatives. Instead of solving for a number, you solve for the unknown function itself. Differential equations are everywhere in science and engineering – they describe how populations grow, how heat spreads, how circuits behave, and how planets orbit.
One of the very first differential equations ever solved was Isaac Newton's law of cooling, published in 1701, which states that an object's temperature changes at a rate proportional to the difference between its own temperature and its surroundings – exactly the dT/dt = −0.1(T−20) form used in the worked examples on this page. The logistic growth equation, dP/dt = rP(1 − P/K), was introduced much later by the Belgian mathematician Pierre François Verhulst in 1838, who noticed that Thomas Malthus's simpler exponential-growth model (used to predict runaway population growth) was unrealistic because real populations eventually run out of resources; Verhulst's equation builds that natural limit directly into the mathematics.
Why Differential Equations?
Many natural processes are described not by what a quantity is, but by how fast it is changing. A differential equation captures that relationship. Solving it reveals the function that describes the full behaviour over time.
Key Terminology
| Term | Meaning | Example |
|---|---|---|
| Order | The highest derivative present | dy/dx = 3y is first order; y″ + y = 0 is second order |
| Degree | The power of the highest derivative | (dy/dx)² = x has degree 2 |
| Ordinary DE | One independent variable | dy/dx = f(x, y) |
| General solution | Family of solutions containing a constant C | y = Cex |
| Particular solution | One specific solution satisfying an initial condition | y = 3ex when y(0) = 3 |
Separable Differential Equations
A first-order DE is separable if it can be written as:
dy/dx = f(x) · g(y)
Separate the variables – get all y terms on one side and all x terms on the other, then integrate both sides.
Worked Examples
Separate variables: dy = 2x dx. Integrate both sides:
∫ dy = ∫ 2x dx → y = x² + C.
General solution: y = x² + C.
Separate: dy/y = 3 dx. Integrate: ln|y| = 3x + C.
Exponentiate: y = Ae3x (where A = eC).
Apply initial condition: y(0) = A = 5. Particular solution: y = 5e3x.
Separate: dy/y = x dx. Integrate: ln|y| = x²/2 + C.
y = Aex²/2. y(0) = A = 2. Particular solution: y = 2ex²/2.
Exponential Growth and Decay
The DE dy/dt = ky models countless real-world situations.
If k > 0: exponential growth (population, compound interest, bacteria).
If k < 0: exponential decay (radioactive decay, cooling, drug concentration).
General solution: y = y0 ekt, where y0 is the value at t = 0.
Doubling time T = 3 h. k = ln(2)/3.
y = 500 e(ln2/3)(9) = 500 e3 ln 2 = 500 × 8 = 4000 bacteria.
Second-Order Linear DEs (Overview)
A second-order linear DE has the form:
ay″ + by′ + cy = 0
Solutions are found using the characteristic equation ar² + br + c = 0. The nature of the roots (real distinct, repeated, or complex) determines the form of the general solution. These equations model oscillations: springs, pendulums, and electrical circuits.
Common Mistakes
| Mistake | Correction |
|---|---|
| Forgetting the constant C in the general solution | Always include C until an initial condition pins it down |
| Separating variables incorrectly | Every y term must move to the dy side; every x term to the dx side |
| Treating ln|y| as ln(y) without considering sign | Use absolute value inside the logarithm; resolve sign when applying the initial condition |
Key Takeaways
- A differential equation relates a function to its derivatives.
- Separable DEs: separate variables then integrate both sides.
- dy/dt = ky gives exponential growth (k > 0) or decay (k < 0).
- Initial conditions turn general solutions into particular solutions.
Practice: Solving Differential Equations
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