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.
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 Questions
- Solve dy/dx = 4x³.
- Solve dy/dx = −2y, given y(0) = 10.
- A radioactive substance decays at 5% per hour. How much remains of a 200 g sample after 10 hours?
- Solve dy/dx = y/x, given y(1) = 3.
- The temperature T of a cooling object satisfies dT/dt = −0.1(T − 20). If T(0) = 80, find T(t) and the temperature after 10 minutes.