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Linearity means the following. Write the equation in the form Lu = 0, where L is an
operator. That is, if v is any function, Lv is a new function. For instance, L = ∂/∂x
is the operator that takes v into its partial derivative v x . In example 2 the operator L is
L = ∂/∂x + y∂/∂y. (Lu = u x + yu y .) The definition we want for linearity is

L(u + v) = Lu + Lv, L(cu) = cLu (1.3)

for any functions u, v and any constant c. Whenever (1.3) holds (for all choices of u, v,
and c), L is called a linear operator. The equation

Lu = 0 (1.4)

is called linear if L is a linear operator. Equation (1.4) is called a homogeneous linear
equation. The equation
Lu = g, (1.5)

where g 6= 0 is a given function of the independent variables, is called an inhomogeneous
linear equation. For instance, the equation

2
2
2
2
(cos xy )u x − y u y = tan(x + y )
is an inhomogeneous linear equation.
As you can easily verify, five of the eight equations above are linear as well as homoge-
neous. Example 5, on the other hand, is not linear because although (u+v) xx = u xx +v xx
and (u + v) tt = u tt + v tt satisfy the property (3), the cubic term does not:

3
2
2
3
3
3
3
(u + v) = u + 3u v + 3uv + v 6= u + v .
The advantage of linearity for the equation Lu = 0 is that if u and v are both solutions,
so is (u + v). If u 1 , . . . , u n are all solutions, so is any linear combinationc
n
X
c 1 u 1 (x) + · · · + c h u n (x) = c j u j (x) (c j = constants).
j=1
(This is sometimes called the superposition principle.) Another consequence of linearity is
that if you add a homogeneous solution [a solution of (1.4)] to an inhomogeneous solution
[a solution of (1.5)], you get an inhomogeneous solution. The mathematical structure
that deals with linear combinations and linear operators is the vector space. We’ll study,
almost exclusively, linear systems with constant coefficients.
For example, the following two equations are both nonlinear:

3
u xx + u yy = u , (1.6)
2
u xx + u yy = |∇u| . (1.7)

Here |∇u| denotes the norm of the gradient of u. While (1.7) is nonlinear, it is still linear
as a function of the highest-order derivative. Such a nonlinearity is called quasilinear. On
the other hand in (1.6) the nonlinearity is only in the unknown function. Such equations
are often called semilinear.

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