Differential equations – further maths revision notes to download as PDF

Starter activity

A bath is filling from a tap at a steady rate, but the plug is not quite in, so water also drains away at a rate that depends on how deep it is. At first the level rises quickly, then more slowly, and eventually it stops rising. Why does it stop? Sketch a graph of depth against time and mark the final depth. Write a sentence in words that describes how the rate of change of the depth depends on the depth. You have just described a differential equation, and in this chapter you learn to solve this kind of equation exactly.

Note: other models

Newton's law of cooling is \(\frac{dT}{dt}=-k(T-T_0)\), with solution \(T=T_0+(T_1-T_0)e^{-kt}\). Mixing problems use \(\frac{dm}{dt}=\text{rate in}-\text{rate out}\), where the rate out depends on the current amount. State the final answer in the context of the problem, with units.

Note: two kinds of solution

A differential equation of order \(n\) has a general solution that contains \(n\) arbitrary constants. Each choice of the constants gives a particular solution, and a set of conditions on \(y\) at a single point, such as the value of \(y\) and the value of \(y'\) at \(x=0\), picks out exactly one of them. It is a common mistake to substitute the initial conditions into the complementary function alone. The conditions apply to the whole of the general solution, which is the complementary function plus the particular integral, so you must find the particular integral first and only then use the conditions.

Note: how to check a solution

You can always check an answer to a differential equation by differentiating it and substituting back into the original equation. For a first order equation this takes only a few lines and is worth doing in an examination if you have spare time. For a second order equation, check that your solution satisfies both initial conditions as well. A solution that does not satisfy the equation is wrong, and the check will almost always reveal where a sign has gone astray.

Applications of differential equations

Almost every law of physics is a differential equation. Newton's second law relates the acceleration of a body to the forces on it, and when the force depends on position or velocity the law becomes a differential equation for the position. Radioactive decay, the cooling of coffee, the charging of a phone battery and the spread of a medicine through the body are all described by first order equations. Pendulums, springs, vibrating bridges, and electrical circuits with coils and capacitors lead to second order equations of the type studied here. Learning to set up such equations from a written description is a very valuable skill, because the algebra of solving them is the easy part once the model is correct.

Why the integrating factor works

The integrating factor can feel like a trick, so it helps to see the idea behind it. The left-hand side of the equation \(\dfrac{dy}{dx}+Py\) looks almost like the result of the product rule, but not quite. If we multiply the whole equation by a suitable function \(I(x)\), then we want \(I\dfrac{dy}{dx}+IPy\) to be exactly the derivative of the product \(Iy\). The product rule says that the derivative of \(Iy\) is \(I\dfrac{dy}{dx}+\dfrac{dI}{dx}y\), so we need \(\dfrac{dI}{dx}=PI\). That is a simple equation whose solution is \(I=e^{\int P\,dx}\). Once the left-hand side is a perfect derivative, we integrate both sides and the problem is solved. You do not need to include a constant of integration when you find \(I\), because any multiple of an integrating factor is also an integrating factor, and the extra constant would cancel at the end.

Reading a model critically

When you have solved a modelling question, always ask whether the answer makes sense. A cooling object cannot become colder than its surroundings, so the temperature must level off at room temperature. A drug concentration cannot become negative. A mass on a spring with damping should lose energy, so its oscillations should die away rather than grow. In an examination you may be asked to comment on the limitations of the model: for example, Newton's law of cooling assumes that the surroundings stay at a constant temperature, and the model for a skydiver assumes that air resistance is proportional to speed, which is not accurate at high speeds. Writing one sensible sentence about this can earn an extra mark.

Comparing the two types of equation

First order and second order equations look different, but they share a common structure. In both cases the general solution is the sum of a part that solves the simpler homogeneous equation, which has zero on the right-hand side, and a particular solution that responds to the forcing term. The homogeneous part usually contains the constants of integration and describes how the system would behave if left alone, for example the natural decay of a temperature or the natural swing of a spring. The particular part describes how the system is pushed by an outside influence. In many practical problems the homogeneous part decays quickly and is called the transient, leaving the particular part as the long-term or steady state behaviour. Recognising this pattern helps you to predict the shape of a solution before you calculate it.

Did you know?

Differential equations were invented by Isaac Newton and Gottfried Leibniz in the seventeenth century, when they developed calculus. Newton used them to show that a single law of gravity explains both the fall of an apple and the orbit of the Moon. Today, the same equations control the design of aircraft, the forecasting of the weather, and the models used to track the spread of diseases. When the pendulum clock was invented by Christiaan Huygens in 1656, its small oscillations were the first example of simple harmonic motion applied to measuring time accurately.