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1.1 What Is a Differential Equation?
Some equations are referred to as nonhomogeneous. They include terms
that do not depend on the unknown u. Typically, (c), (d), and (e) are
nonhomogeneous equations. Furthermore,

u (x)+ u (x)=0

would be the homogeneous counterpart of d). Similarly, the Laplace equa-
tion
u xx (x, y)+ u yy (x, y)=0
is homogeneous, whereas the Poisson equation 3
u xx (x, y)+ u yy (x, y)= f(x, y)
is nonhomogeneous.
An important distinction is between linear and nonlinear equations. In
order to clarify these concepts, it is useful to write the equation in the form
L(u)=0. (1.3)
With this notation, (a) takes the form (1.3) with
L(u)= u (t) − u(t).

Similarly, (j) can be written in the form (1.3) with

L(u)= u tt − u xx + u .
3
Using this notation, we refer to an equation of the form (1.3) as linear if

L(αu + βv)= αL(u)+ βL(v) (1.4)

for any constants α and β and any relevant functions u and v. An equation
2
of the form (1.3) not satisfying (1.4) is nonlinear.
Let us consider (a)above.Wehave

L(u)= u − u,
and thus

2
We have to be a bit careful here in order for the expression L(u) to make sense. For
instance, if we choose

−1 x ≤ 0,
u =
1 x> 0,
then u is not differentiable and it is difficult to interpret L(u). Thus we require a certain
amount of differentiability and apply the criterion only to sufficiently smooth functions.