Applied Maths: Differential Equations (Strand 10) — Higher Level Notes

1. What order means

The order of a differential equation is the highest derivative it contains. ds/dt = 2s is first-order; an equation in d²s/dt² is second-order. At Leaving Cert the work is almost entirely first-order and separable.

2. Separation of variables

dy/dx = f(x)g(y) ⇒ ∫ dy/g(y) = ∫ f(x) dx

Get every y on one side with dy, every x on the other with dx, integrate both sides, and add a single constant. The most common integral you will meet on the left is the logarithmic one, ∫dy/y = ln y.

3. General and particular solutions

Integrating leaves an arbitrary constant, and that is the general solution. Substituting a given condition — an initial speed, a starting population — fixes the constant and gives the particular solution.

A tidy step worth doing every time: when the answer emerges as ln y = f(x) + c, exponentiate to get y = Ae^f(x), where A = e^c. The answer is far easier to interpret in that form, and later parts of the question usually need it.

4. Motion against resistance

The classic application. A body falling under gravity against a resistance proportional to speed satisfies dv/dt = g - kv, which separates and integrates to:

v = (g/k)(1 - e^-kt)

As t grows the exponential dies away and the speed approaches a limit:

v_terminal = g/k

You can also read terminal velocity straight off the equation without solving it at all — it is the speed at which the acceleration becomes zero. Worth knowing as a check.

5. Growth, decay and cooling

All the same equation wearing different clothes. Any quantity whose rate of change is proportional to its current size satisfies dP/dt = kP, giving P = P₀e^kt — with k positive for growth and negative for decay.

Newton's law of cooling is the same idea applied to the difference between an object's temperature and its surroundings. Compound interest is the same idea again in a financial setting.

6. Difference equations or differential equations?

The distinction is simply whether the change happens in steps or continuously. Loan repayments land once a month, so they are a difference equation (Strand 8). A cooling cup of coffee changes at every instant, so it is a differential equation. Some situations can reasonably be modelled either way, and the exam sometimes asks you to say why.