What a differential equation is
A differential equation is an equation that contains the derivative of a dependent variable with respect to an independent variable.
What it means
In an equation like x² − 5x + 6 = 0, the unknown is a number. In a differential equation (DE), the unknown is a function y = f(x). The equation links x, y and the derivatives of y. Here y is the dependent variable and x is the independent variable. If a DE has only one independent variable, it is an ordinary differential equation. Equations with partial derivatives are not in your syllabus, so in this chapter 'DE' always means an ordinary DE.
Is it a differential equation?
| Equation | DE? | Reason |
|---|---|---|
| dy/dx = cos x | Yes | Contains dy/dx |
| x(d²y/dx²) + (dy/dx)² = y | Yes | Contains derivatives of y |
| y = 3x² + 5 | No | No derivative; it only gives y in terms of x |
| x² + y² = 25 | No | An ordinary equation of a circle |
| dP/dt = kP | Yes | P depends on time t |
| d²s/dt² = −9.8 | Yes | Second derivative of s with respect to t |
Where differential equations come from
- Rate of change of any quantity is a derivative: rate of change of P with time is dP/dt.
- 'Proportional to' means 'equal to a constant times': growth proportional to P gives dP/dt = kP.
- The slope of a curve at a point (x, y) is dy/dx, so a rule about the slope is a DE.
- In physics, velocity is ds/dt and acceleration is d²s/dt², so laws of motion are DEs.
Example 1: write the DE for a growing town
The population P of a town grows at a rate proportional to the population at that time. Write this as a differential equation. Name the dependent and independent variables.
- Rate of growth of P with time t = dP/dt.
- 'Proportional to P' means dP/dt = kP for some constant k.
- The town is growing, so k > 0.
- P changes with t, so P is the dependent variable and t is the independent variable.
dP/dt = kP (k > 0). Dependent variable: P. Independent variable: t.
Example 2: a rule about the slope
At every point (x, y) of a curve, the slope of the curve equals the sum of the x-coordinate and twice the y-coordinate. Write the differential equation of the curve.
- The slope of the curve at (x, y) is dy/dx.
- The sum of the x-coordinate and twice the y-coordinate = x + 2y.
- The slope equals this sum, so dy/dx = x + 2y.
dy/dx = x + 2y
After you take a tablet, the amount C of medicine in the blood falls at a rate proportional to C: dC/dt = −kC. Doctors use this kind of DE to decide how often a dose is taken.
1 marks: “Which one is a differential equation: y = sin x + 2, y(dy/dx) = x, x² + y² = 4 or y = mx + c?” y(dy/dx) = x, because it is the only one that contains a derivative of y.
2 marks: “Money in a bank grows continuously so that its rate of increase is 8% of the amount A present. Write the differential equation and name the variables.” Rate of increase = dA/dt. So dA/dt = (8/100)A = 0.08A. A (amount) is dependent; t (time in years) is independent.
Derivative inside = differential equation. No derivative, no DE.