Difference of Linear, Semilinear, and Quasi-Linear Equations

The difference between linear semilinear and quasi-linear PDE is important in the classification of first-order partial differential equations. We compare these three classes by looking at the derivative terms and the coefficients. The comparison depends mainly on two questions. First, do the first partial derivatives appear linearly? Second, do the coefficients depend on the unknown function

u

? For

u=u(x,y)

, the general quasi-linear form is

\begin{equation*} a(x,y,u)u_x+b(x,y,u)u_y=c(x,y,u). \quad \text{(1)} \end{equation*}

The general semilinear form is

\begin{equation*} a(x,y)u_x+b(x,y)u_y=c(x,y,u). \quad \text{(2)} \end{equation*}

The general linear form is

\begin{equation*} a(x,y)u_x+b(x,y)u_y+c(x,y)u=d(x,y). \quad \text{(3)} \end{equation*}

A quasi-linear equation is linear in the first partial derivatives. A semilinear equation is quasi-linear with derivative coefficients independent of

u

. A linear equation is semilinear and is also linear in

u

u

? For

u=u(x,y)

, the general quasi-linear form is

\begin{equation*} a(x,y,u)u_x+b(x,y,u)u_y=c(x,y,u). \quad \text{(1)} \end{equation*}

The general semilinear form is

\begin{equation*} a(x,y)u_x+b(x,y)u_y=c(x,y,u). \quad \text{(2)} \end{equation*}

The general linear form is

\begin{equation*} a(x,y)u_x+b(x,y)u_y+c(x,y)u=d(x,y). \quad \text{(3)} \end{equation*}

A quasi-linear equation is linear in the first partial derivatives. A semilinear equation is quasi-linear with derivative coefficients independent of

u

. A linear equation is semilinear and is also linear in

u

The largest class among these three is the quasi-linear class. The semilinear class is smaller. The linear class is the most restrictive class.

Every linear first-order partial differential equation is semilinear. Every semilinear first-order partial differential equation is quasi-linear. The reverse statements are not always true.

The classes are nested because each smaller class satisfies all the conditions of the larger class. Choose a class and click Show Class Position to see where it sits in the ladder. Notice that linear equations satisfy the semilinear and quasi-linear conditions, while quasi-linear equations need not satisfy the stronger semilinear or linear conditions. This picture helps you remember the direction of inclusion.

For a quasi-linear equation, the coefficients of

u_x

and

u_y

may depend on

u

. For a semilinear equation, these coefficients must not depend on

u

. For a linear equation, the whole equation must be linear in

u

u_x

, and

u_y

The three classes are separated by three checks: how the derivatives appear, whether derivative coefficients contain

u

, and whether the unknown function itself appears linearly. Choose a class and click Compare Conditions to inspect these checks. Notice that all three classes require the first derivatives to appear linearly. The difference comes from stronger restrictions on coefficients and on the appearance of

u

Consider

\begin{equation*} u u_x+yu_y=x. \quad \text{(4)} \end{equation*}

This equation is quasi-linear because

u_x

and

u_y

appear linearly. But it is not semilinear because the coefficient of

u_x

u

Consider

\begin{equation*} xu_x+yu_y=u^2+x. \quad \text{(5)} \end{equation*}

This equation is semilinear. The coefficients of

u_x

and

u_y

are

x

and

y

, and they do not depend on

u

. But the equation is not linear because the term

u^2

appears.

Consider

\begin{equation*} xu_x+yu_y+u=x+y. \quad \text{(6)} \end{equation*}

This equation is linear. The terms

u_x

u_y

, and

u

appear linearly. Their coefficients depend only on

x

and

y

The three examples show the main differences between the classes. Select an example and click Classify Example to reveal the classification and the reason. Notice how the decision changes when a derivative coefficient contains

u

, when a non-derivative term such as

u^2

appears, or when everything involving

u

remains linear. This comparison prepares you for classifying a new equation.

The following list gives the main idea in words. Quasi-linear: first partial derivatives appear linearly, but their coefficients may depend on

u

. Semilinear: first partial derivatives appear linearly, and their coefficients do not depend on

u

. Linear:

u

u_x

, and

u_y

all appear linearly, and their coefficients depend only on the independent variables.

First check whether

u_x

and

u_y

occur linearly. If not, the equation is not quasi-linear. If yes, check whether their coefficients depend on

u

. Finally, check whether

u

itself appears linearly.

Some books use the title Classification of First-Order PDEs. We use Difference Between Linear, Semilinear, and Quasi-Linear PDEs because it directly answers a common student question.

Classify the following equation as linear, semilinear, quasi-linear, or none of these:

\begin{equation*} (1+u)u_x+yu_y=xu. \end{equation*}

The classification test works in a fixed order. First check whether the first derivatives appear linearly, then check whether their coefficients contain

u

, and finally check whether the whole equation is linear in

u

. Choose a sample equation or type your own expression, then click Classify PDE. Use the explanation to see why an equation is linear, semilinear, quasi-linear only, or none of these.

The equation is quasi-linear. It is not semilinear because the coefficient of

u_x

1+u

, which depends on

u

BMLabs Mathematics

BMLabs Mathematics

Difference of Linear, Semilinear, and Quasi-Linear Equations

Difference of Linear, Semilinear, and Quasi-Linear Equations

Class Inclusion Ladder

Condition Comparison Panel

Example Classification Explorer

First Order PDE Classifier

Coordinates and Locus

Invariants under Orthogonal Transformation