15

Ordinary Differential Equations

Ordinary differential equations bring together all the calculus that we have done so far in
the book into one of the most powerful and useful tools of engineering mathematics. In this
chapter we concentrate on the principles of the key methods, rather than the intricate details
of manipulation. It helps to keep in mind the main steps of solving any differential equation:

• identify thetypeof the differential equation by inspecting its form


• choose amethodof solution appropriate to the type


• solvethe equation, including any extra conditions


• checkthe solution by substituting back into the differential equation
  The last point may seem tedious at times, but not only does it give you greater confidence
  in the solution, but it gives you essential practice in differentiation and other areas of
  mathematics.

Prerequisites

It will be helpful if you know something about:

• elementary algebra such as partial fractions (62

➤

)

• the exponential function and its properties (Chapter 4)


• differentiation (Chapter 8)


• integration (Chapter 9)


• sines and cosines and their properties (Chapter 6)


• exponential form of a complex number (361

➤

)

• solving simultaneous equations (391

➤

)

In particular, it will be of greatest benefit if you know and fully understand the

following results:

Ify=Aekx,whereAandkare constants then

dy

dx

=kAekx=ky

Ify=AcoskxorAsinkxthen

d^2 y

dx^2

=−k^2 y

These two results are fundamental to differential equations and represent protype

equations of first and second order to which we already know the solutions.