C ha pter 2
L inear First-Or der PDEs
The general first-order partial differential equation (PDE) for a two-variable function, denoted as
u=u(x; y), can be expressed in the form:
F (x; y; u; u
x
; u
y
) = 0:
Here, u
x
and u
y
repre sent the partial derivatives of u with respe ct to x and y, respectively. The
function F establishe s a functional relationship between the function u, its partial derivatives, a nd
the independent variables x and y. The general first-order partial differential equation f or a function
u=u(x; : : : ; x) o f n independent variables, denoted as x; : : : ; x, can be represented as:
F (x; u; ru)=0;
Here, ru is a vector denoted as ru= (@u; : : : ; @u), which comprises the partial derivatives of u
with respect to each independent variable x; : : : ; x.
Definition 2.1. A classical solution of the equation F (x; u; ru)=0, for x2R, ru= (@u; :::; @u),
is a smooth function u=u(x) defined on an open set R such that F (x; u(x); ru(x))=0, is an
identity for all x 2.
For example, it is possible to verify that functions of the form u =h(x
2
+ y
2
) for arbitrary smooth
functions h is the classical solution of the equation
yu
x
¡xu
y
=0:
For example, the function u = x
2
+ y
2
is a classical solution to the equation for all (x; y) 2R
2
, while
u = x
2
+ y
2
p
is a solution only on R
2
¡f(0; 0)g. The gr aph of these two solution shown below.
Observe that the graph of a classica l solution of a first-order PDE in two variable x; y is a smooth
surface:
1
2.1 Cl assification of first-order PDEs
In this book, we will exclusively focus on the study of first-order PDEs falling within the categories
of linear, semi-linear, and quasi-linear. The subject of fully nonline ar equations will be introduced
in a separate book dedicated to that topic.
Linear equations. The general form of a linear first-order PDE for a function u=u(x), x =
(x
1
; : : : ; x
n
) is given by:
X
j=1
n
v
j
(x) @
j
u(x) + v
0
(x) u = r(x);
for some (usually) continuous functions v(x) and r(x).
Semi-linear equations. A semi-linear equation is characterized by the general form:
X
j=1
n
v
j
(x) @
j
u(x) = r(x; u);
The difference between a linear and semi-linear equation is that a semi-linear equation can
be nonlinear with respect to u (and not with the partial deriva tives u
x
and u
y
).
Quasi-linear equations. A quasi-linear equation assumes the general form:
X
j=1
n
v
j
(x; u) @
j
u(x) = r(x; u);
The difference between a quasi-linear and semi-linear equation is that in the former case, the
coefficients of partial derivatives are function of u a s well.
Fully-nonlinear equations. A fully nonlinear equation is an equation where one or all of the
partial derivatives are nonlinear. For example, the equation:
ju
x
j² + ju
y
j² =1;
is a fully nonlinear first-order equatio n for u=u(x; y). We wont study this type of equation
in this book.
Exercise 2.1. Classify the following rst-order eq uations
a) u
x
+ u
y
= e
x
u
b) xu
x
+ yu
y
= e
u
c) u
x
+ (u
2
)
x
= 1 + u
d) u
x
u
y
+ uu
z
= 1
Exercise 2.2. Consider t he linear equation
u
x
+ u
y
= ¡u:
Verify that every fun ction of the forms u = f(y ¡ x) e
¡x
, u = f (y ¡x) e
¡y
satisfies the equation, where f is a
smooth arbitrary function.
Exercise 2.3. Consider t he following quas i- linear equation
u
x
+ uu
y
= 0:
a) Verify tha t every function of the implicit form u = f(y ¡ux) solves the equation.
b) What is t he exp licit solution if we know u satis es the con dition u = 2y + 1 along the y-axis ?
Exercise 2.4. Verify that the function u = 1 ¡ x
2
+ y
2
p
is a solution t o the fully nonlinear equation
ju
x
j
2
+ ju
y
j
2
= 1:
What is the domain of u if it considered as a c lassical solution? The solu tion satisfies th e auxiliary condition
u = 1 on the unit circle x
2
+ y
2
= 1. Note that u = x
2
+ y
2
p
¡1 solves the PDE and the auxiliary condition too.
2 Li near First-Order PDEs
2.2 Characteristic method and OD Es along cu rves
The characteristic method is a powerful technique for solving first-order partial differential equa-
tions, and it is especially useful for semi-linear and quasi-linear equations. By using this method,
one can derive the general solution to such equations. Furthermore, the characteristic method has
a geometric interpretation that can be illustrated through the Cauchy problem.
2.2.1 Introductory remark: ODE along a curve
In our study of ordinary differential equations (ODEs), we explore d equations of the form:
du
dx
= f (x; u):
Here, u = u(x) represents a single-variable function. Geometrically, we interpret the x-variable as the
x-axis in the standard direction. The solution to this equation consists of a one-parameter family of
functions u = u(x; c), where c 2R, such that for any x within the domain of u, the following relation
holds:
d
dx
u(x; c) = f(x; u(x; c)):
Now, let's shift our focus to a parametric curve γ(t) in the xy-plane. An ordinary differential
equation (first-order) along γ(t) takes the form:
du γ
dt
= f (t; u γ):
Here, uγ is defined at any t as (uγ)(t)= u(γ(t)). If we denote w as uγ, we arrive at the equation:
dw
dt
= f (t; w):
γ(t) = (x(t); y(t))
x
dw
dt
= f(t; w)
y
For instance, consider γ(t) given by γ = (t; t
2
), a parabola in the xy-plane along which the
differential equati on:
dw
dt
= w;
is defined. Suppose u at the point (1 ; 0) is 1, corresponding to γ(0). Then, we obtain w as:
w(t) = u(γ(t)) = e
t
:
2.2 Characteristic method and ODEs along curves 3
Remark 2.2. Solving differential equations along curves can sometimes result in non-valid solu-
tions. For instance , let's consider the circle C represented by the parametric curve:
γ(t) = (cos(t); sin(t));
for t in the interval [0; 2π), along with the initial value problem:
dw
dt
= w; w(0) = 1;
defined for w(t) = u(γ(t)). The solution to this equation is w(t) = e
t
for t 2[0; 2π). However, the
function u is not continuous on the circle because:
u(γ(0)) = u(1; 0) = 1;
and
u
lim
t!2π
γ(t)
= u(γ(0)) = 1 =/ lim
t!2π
u(γ(t))
Exercise 2.5. Consider the famil y of د fy = x
2
+ c; c 2Rg for parameter c 2R. Consider th e ord inary differential
eq uation
du
dx
= 0;
alo ng all curves i n this family. Show that u on the xy-plane can be descri bed by u = h(y ¡ x
2
) where h is an
ar bitrary smooth function.
2.2.2 A simple type of e quations
Let's begin with the following simple equation:
u
x
+ v(x; y) u
y
= 0;