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(x₁; : : : ; xₙ) o f n independent variables, denoted as x₁; : : : ; xₙ, can be represented as:

F (x; u; ru)=0;

Here, ru is a vector denoted as ru= (@₁u; : : : ; @ₙu), which comprises the partial derivatives of u

with respect to each independent variable x₁; : : : ; xₙ.

Deﬁnition 2.1. A classical solution of the equation F (x; u; ru)=0, for x2Rⁿ, ru= (@₁u; :::; @ₙu),

is a smooth function u=u(x) deﬁned on an open set Ω⊂Rⁿ such that F (x; u(x); ru(x))=0, is an

identity for all x 2Ω.

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 vⱼ(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 ﬁrst-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 ﬁes 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, u◦γ is deﬁned at any t as (u◦γ)(t)= u(γ(t)). If we denote w as u◦γ, 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;