Handbook of mathematics for engineers and scienteists part 185 pptx

1256 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS
9.
∂w
∂x
+

xf(w) + yg(w) + h(w)

∂w
∂y
=0.
General solution: y +
xf(w)+h(w)
g(w)
+
f(w)
g
2
(w)
=exp

g(w)x

Φ(w).
10.
∂w
∂x
+ f(x)g(y)h(w)
∂w
∂y
=0.

General solution:

dy
g(y)
– h(w)

f(x) dx = Φ(w).
T7.2.3. Equations of the Form
∂w
∂x
+ f(x, y, w)
∂w
∂y
= g(x, y, w)
 In the solutions of equations T7.2.3.1–T7.2.3.11, Φ(z) is an arbitrary composite function
whose argument z can depend on x, y, and w.
1.
∂w
∂x
+ aw
∂w
∂y
= f (x).
General solution:
y = ax

w – F(x)

+ a


F (x) dx + Φ

w – F(x)

,whereF (x)=

f(x) dx.
2.
∂w
∂x
+ aw
∂w
∂y
= f (y).
General solution:
x =

y
y
0
dz
√
2aF (z)–2au
+ Φ(u), where F (y)=

f(y) dy, u = F (y)–
1
2
aw
2

.
3.
∂w
∂x
+

aw + f(x)

∂w
∂y
= g(x).
General solution:
y = ax

w – G(x)

+ a

G(x) dx + F(x)+Φ

w – G(x)

,
where
F (x)=

f(x) dx, G(x)=

g(x) dx.
4.

∂w
∂x
+ f(w)
∂w
∂y
= g(x).
General solution: y =

x
x
0
f

G(t)–G(x)+w

dt +Φ

w – G(x)

,whereG(x)=

g(x) dx.
T7.2. QUASILINEAR EQUATIONS 1257
5.
∂w
∂x
+ f(w)
∂w
∂y
= g(y).

General solution:
x =

y
y
0
ψ

G(t)–G(y)+F (w)

dt + Φ

F (w)–G(y)

,
where G(y)=

g(y) dy and F (w)=

f(w) dw. The function ψ = ψ(z)isdefined
parametrically by ψ =
1
f(w)
, z = F (w).
6.
∂w
∂x
+ f(w)
∂w
∂y

= g(w).
General solution: y =

f(w)
g(w)
dw + Φ

x –

dw
g(w)

.
7.
∂w
∂x
+

f(w) + g(x)

∂w
∂y
= h(x).
General solution:
y =

x
x
0
f


H(t)–H(x)+w

dt + G(x)+Φ

w – H(x)

,
where
G(x)=

g(x) dx, H(x)=

h(x) dx.
8.
∂w
∂x
+

f(w) + g(x)

∂w
∂y
= h(w).
General solution:
y =

f(w)
h(w)
dw +


w
w
0
g

H(t)–H(w)+x

h(t)
dt + Φ

x – H(w)

,whereH(x)=

dw
h(w)
.
9.
∂w
∂x
+

f(w) + yg(x)

∂w
∂y
= h(x).
General solution:
yG(x)–


G(x)f

H(t)–H(x)+w

dx = Φ

w – H(x)

,
where G(x)=exp

–

g(x) dx

and H(x)=

h(x) dx.
10.
∂w
∂x
+ f(x, w)
∂w
∂y
= g(x).
General solution: y =

x
x

0
f

t, G(t)–G(x)+w

dt+Φ

w–G(x)

,whereG(x)=

g(x) dx.
1258 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS
11.
∂w
∂x
+ f(x, w)
∂w
∂y
= g(w).
General solution: y =

w
w
0
f

G(t)–G(w)+x, t

g(t)

dt+Φ

x–G(w)

,whereG(w)=

dw
g(w)
.
T7.3. Nonlinear Equations
T7.3.1. Equations Quadratic in One Derivative
 In this subsection, only complete integrals are presented. In order to construct the
corresponding general solution, one should use the formulas of Subsection 13.2.1.
1.
∂w
∂x
+ a

∂w
∂y

2
= by.
This equation governs the free vertical drop of a point body near the Earth’s surface (y is the
vertical coordinate measured downward, x time, m =
1
2a
the mass of the body, and g = 2ab
the gravitational acceleration).
Complete integral: w =–C

1
x
2a
3b

by + C
1
a

3/2
+ C
2
.
2.
∂w
∂x
+ a

∂w
∂y

2
+ by
2
=0.
This equation governs free oscillations of a point body of mass m = 1/(2a) in an elastic field
with elastic coefficient k = 2b (x is time and y is the displacement from the equilibrium).
Complete integral: w =–C
1
x + C

2


C
1
– by
2
a
dx + C
2
.
3.
∂w
∂x
+ a

∂w
∂y

2
= f (x) + g(y).
Complete integral: w =–C
1
x +

f(x) dx +


g(y)+C
1

a
dy + C
2
.
4.
∂w
∂x
+ a

∂w
∂y

2
= f (x)y + g(x).
Complete integral:
w = ϕ(x)y +


g(x)–aϕ
2
(x)

dx + C
1
,whereϕ(x)=

f(x) dx + C
2
.
5.

∂w
∂x
+ a

∂w
∂y

2
= f (x)w + g(x).
Complete integral:
w = F (x)(C
1
+ C
2
y)+F (x)


g(x)–aC
2
2
F
2
(x)

dx
F (x)
,whereF(x)=exp


f(x) dx


.
T7.3. NONLINEAR EQUATIONS 1259
6.
∂w
∂x
– f(w)

∂w
∂y

2
=0.
Complete integral in implicit form:

f(w) dw = C
2
1
x + C
1
y + C
2
.
7. f
1
(x)
∂w
∂x
+ f
2

(y)

∂w
∂y

2
= g
1
(x) + g
2
(y).
Complete integral: w =

g
1
(x)–C
1
f
1
(x)
dx +


g
2
(y)+C
1
f
2
(y)

dy + C
2
.
8.
∂w
∂x
+ a

∂w
∂y

2
+ b
∂w
∂y
= f (x) + g(y).
Complete integral: w =–C
1
x + C
2
+

f(x) dx –
b
2a
y
1
2a



4ag(y)+b
2
+ 4aC
1
dy.
9.
∂w
∂x
+ a

∂w
∂y

2
+ b
∂w
∂y
= f (x)y + g(x).
Complete integral:
w = ϕ(x)y +


g(x)–aϕ
2
(x)–bϕ(x)

dx + C
1
,whereϕ(x)=


f(x) dx + C
2
.
10.
∂w
∂x
+ a

∂w
∂y

2
+ b
∂w
∂y
= f (x)w + g(x).
Complete integral:
w =(C
1
y+C
2
)F (x)+F (x)


g(x)–aC
2
1
F
2
(x)–bC

1
F (x)

dx
F (x)
, F(x)=exp


f(x) dx

.
T7.3.2. Equations Quadratic in Two Derivatives
1. a

∂w
∂x

2
+ b

∂w
∂y

2
= c.
For a = b,thisisadifferential equation of light rays.
Complete integral: w = C
1
x + C
2

y + C
3
,whereaC
2
1
+ bC
2
2
= c.
An alternative form of the complete integral:
w
2
c
=
(x – C
1
)
2
a
+
(y – C
2
)
2
b
.
2.

∂w
∂x


2
+

∂w
∂y

2
= a –2by.
This equation governs parabolic motion of a point mass in vacuum (the coordinate x is
measured along the Earth’s surface, the coordinate y is measured vertically upward from
the Earth’s surface, and a is the gravitational acceleration).
Complete integral: w = C
1
x
1
3b
(a – C
2
1
– 2by)
3/2
+ C
2
.
1260 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS
3.

∂w
∂x


2
+

∂w
∂y

2
=
a

x
2
+ y
2
+ b.
This equation arises from the solution of the two-body problem in celestial mechanics.
Complete integral:
w =


b +
a
r
–
C
2
1
r
2

dr + C
1
arctan
y
x
+ C
2
,wherer =

x
2
+ y
2
.
4.

∂w
∂x

2
+

∂w
∂y

2
= f (x).
Complete integral: w = C
1
y + C

2


f(x)–C
2
1
dx.
5.

∂w
∂x

2
+

∂w
∂y

2
= f (x) + g(y).
Complete integral: w =


f(x)+C
1
dx


g
2

(y)–C
1
dy + C
2
. The signs before
each of the integrals can be chosen independently of each other.
6.

∂w
∂x

2
+

∂w
∂y

2
= f (x
2
+ y
2
).
Hamilton’s equation for the plane motion of a point mass under the action of a central force.
Complete integral:
w = C
1
arctan
x
y

+ C
2
1
2


zf(z)–C
2
1
dz
z
, z = x
2
+ y
2
.
7.

∂w
∂x

2
+

∂w
∂y

2
= f (w).
Complete integral in implicit form:


dw

f(w)
=

(x + C
1
)
2
+(y + C
2
)
2
.
8.

∂w
∂x

2
+
1
x
2

∂w
∂y

2

= f (x).
This equation governs the plane motion of a point mass in a central force field, with x and y
being polar coordinates.
Complete integral: w = C
1
y


f(x)–
C
2
1
x
2
dx + C
2
.
9.

∂w
∂x

2
+ f(x)

∂w
∂y

2
= g(x).

Complete integral: w = C
1
y + C
2
+


g(x)–C
2
1
f(x) dx.
T7.3. NONLINEAR EQUATIONS 1261
10.

∂w
∂x

2
+ f(y)

∂w
∂y

2
= g(y).
Complete integral: w = C
1
x + C
2
+



g(y)–C
2
1
f(y)
dy.
11.

∂w
∂x

2
+ f(w)

∂w
∂y

2
= g(w).
Complete integral in implicit form:


C
2
1
+ C
2
2
f(w)

g(w)
dw = C
1
x + C
2
y + C
3
.
One of the constants C
1
or C
2
can be set equal to 1.
12. f
1
(x)

∂w
∂x

2
+ f
2
(y)

∂w
∂y

2
= g

1
(x) + g
2
(y).
A separable equation. This equation is encountered in differential geometry in studying
geodesic lines of Liouville surfaces. Complete integral:
w =


g
1
(x)+C
1
f
1
(x)
dx


g
2
(y)–C
1
f
2
(y)
dy + C
2
.
The signs before each of the integrals can be chosen independently of each other.

T7.3.3. Equations with Arbitrary Nonlinearities in Derivatives
1.
∂w
∂x
+ f

∂w
∂y

=0.
This equation is encountered in optimal control and differential games.
1
◦
. Complete integral: w = C
1
y – f(C
1
)x + C
2
.
2
◦
. On differentiating the equation with respect to y, we arrive at a quasilinear equation of
the form T7.2.2.3:
∂u
∂x
+ f

(u)
∂u

∂y
= 0, u =
∂w
∂y
,
which is discussed in detail in Subsection 13.1.3.
3
◦
. The solution of the Cauchy problem with the initial condition w(0, y)=ϕ(y) can be
written in parametric form as
y = f

(ζ)x + ξ, w =

ζf

(ζ)–f (ζ)

x + ϕ(ξ), where ζ = ϕ

(ξ).
See also Examples 1 and 2 in Subsection 13.2.3.
2.
∂w
∂x
+ f

∂w
∂y


= g(x).
Complete integral: w = C
1
y – f(C
1
)x +

g(x) dx + C
2
.
1262 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS
3.
∂w
∂x
+ f

∂w
∂y

= g(x)y + h(x).
Complete integral:
w = ϕ(x)y +


h(x)–f

ϕ(x)

dx + C
1

,whereϕ(x)=

g(x) dx + C
2
.
4.
∂w
∂x
+ f

∂w
∂y

= g(x)w + h(x).
Complete integral:
w =(C
1
y + C
2
)ϕ(x)+ϕ(x)


h(x)–f(C
1
ϕ(x))

dx
ϕ(x)
,whereϕ(x)=exp



g(x) dx

.
5.
∂w
∂x
– F

x,
∂w
∂y

=0.
Complete integral: w =

F (x, C
1
) dx + C
1
y + C
2
.
6.
∂w
∂x
+ F

x,
∂w

∂y

= aw.
Complete integral: w = e
ax
(C
1
y + C
2
)–e
ax

e
–ax
F (x, C
1
e
ax
) dx.
7.
∂w
∂x
+ F

x,
∂w
∂y

= g(x)w.
Complete integral:

w = ϕ(x)(C
1
y + C
2
)–ϕ(x)

F

x, C
1
ϕ(x)

dx
ϕ(x)
,whereϕ(x)=exp


g(x) dx

.
8. F

∂w
∂x
,
∂w
∂y

=0.
Complete integral:

w = C
1
x + C
2
y + C
3
,
where C
1
and C
3
are arbitrary constants and the constant C
2
is related toC
1
by F (C
1
, C
2
)=0.
9. w = x
∂w
∂x
+ y
∂w
∂y
+ F

∂w
∂x

,
∂w
∂y

.
Clairaut’s equation. Complete integral: w = C
1
x + C
2
y + F (C
1
, C
2
).