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1. Setting the Scene
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= α(u − u)+ β(v − v)

= αL(u)+ βL(v),
for any constants α and β and any differentiable functions u and v. So this
equation is linear. But if we consider (j), we have
L(u)= u tt − u xx + u ,
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and thus L(αu + βv)= αu + βv − αu − βv
L(u + v)= u tt − u xx + v tt − v xx +(u + v) ,
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which is not equal to L(u)+L(v) for all functions u and v since, in general,
(u + v) = u + v .
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3
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So the equation (j) is nonlinear. It is a straightforward exercise to show
that also (c), (d), (e), (f), (g), (h), (l) and (m) are linear, whereas (b), (i)
and (k), in addition to (j), are nonlinear.
1.2 The Solution and Its Properties
In the previous section we introduced such notions as linear, nonlinear,
order, ordinary differential equations, partial differential equations, and
homogeneous and nonhomogeneous equations. All these terms can be used
to characterize an equation simply by its appearance. In this section we will
discuss some properties related to the solution of a differential equation.
1.2.1 An Ordinary Differential Equation

Let us consider a prototypical ordinary differential equation,
u (t)= −u(t) (1.5)

equipped with an initial condition
u(0) = u 0 . (1.6)

Here u 0 is a given number. Problems of this type are carefully analyzed in
introductory courses and we shall therefore not dwell on this subject. The
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3 Boyce and DiPrima [3] and Braun [5] are excellent introductions to ordinary differ-
ential equations. If you have not taken an introductory course in this subject, you will
find either book a useful reference.