where the coefficientsa, b, andc, in general, are functions ofxandy and d(x, y) is a given function. Unless stated otherwise, these functions are assumed to be continuously differentiable. Equations of the form (2.2.12) are calledhomogeneous ifd(x, y)≡0 ornonhomogeneous ifd(x, y)= 0.
Obviously, linear equations are a special kind of the quasi-linear equa- tion (2.2.4) if a, b are independent of u and c is a linear function in u.
Similarly, semilinear equation (2.2.8) reduces to a linear equation if c is linear inu.
Examples of linear equations are
xux+yuy−nu= 0, (2.2.13) nux+ (x+y)uy−u=ex, (2.2.14) yux+xuy =xy, (2.2.15) (y−z)ux+ (z−x)uy+ (x−y)uz= 0. (2.2.16) An equation which isnot linear is often called anonlinear equation. So, first-order equations are often classified as linear and nonlinear.
2.3 Construction of a First-Order Equation
We consider a system of geometrical surfaces described by the equation f(x, y, z, a, b) = 0, (2.3.1) whereaandbare arbitrary parameters. We differentiate (2.3.1) with respect toxandyto obtain
fx+p fz= 0, fy+q fz= 0, (2.3.2) wherep=∂x∂z andq= ∂z∂y.
The set of three equations (2.3.1) and (2.3.2) involves two arbitrary parameters a and b. In general, these two parameters can be eliminated from this set to obtain a first-order equation of the form
F(x, y, z, p, q) = 0. (2.3.3) Thus the system of surfaces (2.3.1) gives rise to a first-order partial dif- ferential equation (2.3.3). In other words, an equation of the form (2.3.1) containing two arbitrary parameters is called acomplete solutionor acom- plete integralof equation (2.3.3). Its role is somewhat similar to that of a general solution for the case of an ordinary differential equation.
On the other hand, any relationship of the form
f(φ, ψ) = 0, (2.3.4)
which involves an arbitrary functionf of two known functionsφ=φ(x, y, z) andψ=ψ(x, y, z) and provides a solution of a first-order partial differential equation is called a general solution or general integral of this equation.
Clearly, the general solution of a first-order partial differential equation depends on an arbitrary function. This is in striking contrast to the situation for ordinary differential equations where the general solution of a first- order ordinary differential equation depends on one arbitrary constant. The general solution of a partial differential equation can be obtained from its complete integral. We obtain the general solution of (2.3.3) from its complete integral (2.3.1) as follows.
First, we prescribe the second parameter b as an arbitrary function of the first parameter a in the complete solution (2.3.1) of (2.3.3), that is, b = b(a). We then consider the envelope of the one-parameter family of solutions so defined. This envelope is represented by the two simultaneous equations
f(x, y, z, a, b(a)) = 0, (2.3.5) fa(x, y, z, a, b(a)) +fb(x, y, z, b(a))b′(a) = 0, (2.3.6) where the second equation (2.3.6) is obtained from the first equation (2.3.5) by partial differentiation with respect toa. In principle, equation (2.3.5) can be solved fora=a(x, y, z) as a function ofx,y, andz. We substitute this result back in (2.3.5) to obtain
f{x, y, z, a(x, y, z), b(a(x, y, z))}= 0, (2.3.7) where b is an arbitrary function. Indeed, the two equations (2.3.5) and (2.3.6) together define the general solution of (2.3.3). When a definiteb(a) is prescribed, we obtain a particular solution from the general solution.
Since the general solution depends on an arbitrary function, there are in- finitely many solutions. In practice, only one solution satisfying prescribed conditions is required for a physical problem. Such a solution may be called aparticular solution.
In addition to the general and particular solutions of (2.3.3), if the enve- lope of the two-parameter system (2.3.1) of surfaces exists, it also represents a solution of the given equation (2.3.3); the envelope is called thesingular solutionof equation (2.3.3). The singular solution can easily be constructed from the complete solution (2.3.1) representing a two-parameter family of surfaces. The envelope of this family is given by the system of three equa- tions
f(x, y, z, a, b) = 0, fa(x, y, z, a, b) = 0, fb(x, y, z, a, b) = 0. (2.3.8) In general, it is possible to eliminate a and b from (2.3.8) to obtain the equation of the envelope which gives the singular solution. It may be pointed out that the singular solutioncannot be obtained from the general
2.3 Construction of a First-Order Equation 31 solution. Its nature is similar to that of the singular solution of a first-order ordinary differential equation.
Finally, it is important to note that solutions of a partial differential equation are expected to be represented by smooth functions. A function is calledsmoothif all of its derivatives exist and are continuous. However, in general, solutions are not always smooth. A solution which is not ev- erywhere differentiable is called aweak solution. The most common weak solution is the one that has discontinuities in its first partial derivatives across a curve, so that the solution can be represented by shock waves as surfaces of discontinuity. In the case of a first-order partial differential equa- tion, there are discontinuous solutions wherezitself andnotmerelyp=∂z∂x andq= ∂z∂y are discontinuous. In fact, this kind of discontinuity is usually known as ashock wave. An important feature of quasi-linear and nonlinear partial differential equations is that their solutions may develop disconti- nuities as they move away from the initial state. We close this section by considering some examples.
Example 2.3.1.Show that a family of spheres
x2+y2+ (z−c)2=r2, (2.3.9) satisfies the first-order linear partial differential equation
yp−xq= 0. (2.3.10)
Differentiating the equation (2.3.9) with respect toxandy gives x+p(z−c) = 0 and y+q(z−c) = 0.
Eliminating the arbitrary constant c from these equations, we obtain the first-order, partial differential equation
yp−xq= 0.
Example 2.3.2.Show that the family of spheres
(x−a)2+ (y−b)2+z2=r2 (2.3.11) satisfies the first-order, nonlinear, partial differential equation
z2
p2+q2+ 1 =r2. (2.3.12) We differentiate the equation of the family of spheres with respect tox andy to obtain
(x−a) +z p= 0, (y−b) +z q= 0.
Eliminating the two arbitrary constants aandb, we find the nonlinear partial differential equation
z2
p2+q2+ 1 =r2.
All surfaces of revolution with thez-axis as the axis of symmetry satisfy the equation
z=f
x2+y2 , (2.3.13)
where f is an arbitrary function. Writing u=x2+y2 and differentiating (2.3.13) with respect toxandy, respectively, we obtain
p= 2x f′(u), q= 2y f′(u).
Eliminating the arbitrary function f(u) from these results, we find the equation
yp−xq= 0.
Theorem 2.3.1.If φ=φ(x, y, z) andψ =ψ(x, y, z) are two given func- tions ofx,y, andzand iff(φ, ψ) = 0, wheref is an arbitrary function ofφ andψ, thenz=z(x, y) satisfies a first-order, partial differential equation
p ∂(φ, ψ)
∂(y, z) +q ∂(φ, ψ)
∂(z, x) = ∂(φ, ψ)
∂(x, y), (2.3.14) where
∂(φ, ψ)
∂(x, y) = φx φy
ψxψy
. (2.3.15)
Proof. We differentiatef(φ, ψ) = 0 with respect toxandyrespectively to obtain the following equations:
∂f
∂φ ∂φ
∂x +p∂φ
∂z
+∂f
∂ψ ∂ψ
∂x +p∂ψ
∂z
= 0, (2.3.16)
∂f
∂φ ∂φ
∂y +q∂φ
∂z
+∂f
∂ψ ∂ψ
∂y +q∂ψ
∂z
= 0. (2.3.17) Nontrivial solutions for ∂f∂φ and ∂f∂ψ can be found if the determinant of the coefficients of these equations vanishes, that is,
φx+pφz ψx+pψz
φy+qφz ψy+qψz
= 0. (2.3.18)
Expanding this determinant gives the first-order, quasi-linear equation (2.3.14).
2.4 Geometrical Interpretation of a First-Order Equation 33