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1
Setting the Scene

You are embarking on a journey in a jungle called Partial Differential Equa-
tions. Like any other jungle, it is a wonderful place with interesting sights
all around, but there are also certain dangerous spots. On your journey,
you will need some guidelines and tools, which we will start developing in
this introductory chapter.

1.1 What Is a Differential Equation?

The field of differential equations is very rich and contains a large vari-
ety of different species. However, there is one basic feature common to all
problems defined by a differential equation: the equation relates a function
to its derivatives in such a way that the function itself can be determined.
This is actually quite different from an algebraic equation, say

x − 2x + 1=0,
2
whose solution is usually a number. On the other hand, a prototypical
differential equation is given by

u (t)= u(t).
The solution of this equation is given by the function

t
u(t)= ce ,