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 = 21a the mass of the body, and g =2ab the
Trang 11256 FIRST-ORDERPARTIALDIFFERENTIALEQUATIONS
∂x +
xf (w) + yg(w) + h(w) ∂w
∂y = 0.
General solution: y + xf (w) + h(w)
g (w) +
f (w)
g2(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.
∂x + aw ∂w
∂y = f (x).
General solution:
y = ax
w – F (x)
+ a
F (x) dx +Φ w – F (x)
, where F (x) =
f (x) dx.
∂x + aw ∂w
∂y = f (y).
General solution:
x=
y
y0
dz
√
2aF (z) –2au +Φ(u), where F (y) =
f (y) dy, u = F (y) – 1
2aw2.
∂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.
∂x + f (w) ∂w
∂y = g(x).
General solution: y =
x
x0
f G (t) – G(x) + w
dt+Φ w – G(x)
, where G(x) =
g (x) dx.
Trang 25. ∂w
∂x + f (w) ∂w
∂y = g(y).
General solution:
x=
y
y0
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) is defined parametrically by ψ = 1
f (w) , z = F (w).
∂x + f (w) ∂w
∂y = g(w).
General solution: y =
f (w)
g (w) dw+Φ
x–
dw
g (w)
∂x +
f (w) + g(x) ∂w
∂y = h(x).
General solution:
y=
x
x0
f H (t) – H(x) + w
dt + G(x) +Φ w – H(x)
, where
G (x) =
g (x) dx, H (x) =
h (x) dx.
∂x +
f (w) + g(x) ∂w
∂y = h(w).
General solution:
y=
f (w)
h (w) dw+
w
w0
g H (t) – H(w) + x
h (t) dt+Φ x – H(w)
, where H(x) =
dw
h (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
x0
f t , G(t)–G(x)+w
dt+Φ w –G(x)
, where G(x) =
g (x) dx.
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11. ∂w
∂x + f (x, w) ∂w
∂y = g(w).
General solution: y =
w
w0
f G (t) – G(w) + x, t
g (t) dt+Φ x –G(w)
, where G(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.
∂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 = 21a the mass of the body, and g =2ab
the gravitational acceleration)
Complete integral: w = –C1x 2a
3b
by + C1 a
3 2
+ C2
∂x + a
∂w
∂y
2
+ by2 = 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 = –C1x + C2 C1– by2
a dx + C2
∂x + a
∂w
∂y
2
= f (x) + g(y).
Complete integral: w = –C1x+
f (x) dx + g (y) + C1
a dy + C2
∂x + a
∂w
∂y
2
= f (x)y + g(x).
Complete integral:
w = ϕ(x)y +
g (x) – aϕ2(x)
dx + C1, where ϕ(x) =
f (x) dx + C2
∂x + a
∂w
∂y
2
= f (x)w + g(x).
Complete integral:
w = F (x)(C1+ C2y ) + F (x)
g (x) – aC22F2(x) dx
F (x) , where F (x) = exp
f (x) dx
Trang 4
6. ∂w
∂x – f (w)
∂w
∂y
2
= 0.
Complete integral in implicit form:
f (w) dw = C12x + C1y + C2
7. f1(x) ∂w
∂x + f2(y)
∂w
∂y
2
= g1(x) + g2(y).
Complete integral: w =
g1(x) – C1
f1(x) dx+
!
g2(y) + C1
f2(y) dy + C2.
∂x + a
∂w
∂y
2
+ b ∂w
∂y = f (x) + g(y).
Complete integral: w = –C1x + C2+
f (x) dx – b
2a y
1
2a
4ag (y) + b2+4aC1dy
∂x + a
∂w
∂y
2
+ b ∂w
∂y = f (x)y + g(x).
Complete integral:
w = ϕ(x)y +
g (x) – aϕ2(x) – bϕ(x)
dx + C1, where ϕ(x) =
f (x) dx + C2
10. ∂w
∂x + a
∂w
∂y
2
+ b ∂w
∂y = f (x)w + g(x).
Complete integral:
w = (C1y +C2)F (x)+F (x)
g (x)–aC12F2(x)–bC
1F (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, this is a differential equation of light rays.
Complete integral: w = C1x + C2y + C3, where aC12+ bC22= c.
An alternative form of the complete integral: w
2
c = (x – C1)
2
a + (y – C2)
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 = C1x 1
3b (a – C12–2by)3 2+ C2
Trang 51260 FIRST-ORDERPARTIALDIFFERENTIALEQUATIONS
3.
∂w
∂x
2
+
∂w
∂y
2
x2+ y2 + b.
This equation arises from the solution of the two-body problem in celestial mechanics Complete integral:
w=
!
b+ a
r – C
2 1
r2 dr + C1arctan
y
x + C2, where r =
x2+ y2.
4.
∂w
∂x
2
+
∂w
∂y
2
= f (x).
Complete integral: w = C1y + C2
f (x) – C12dx
5.
∂w
∂x
2
+
∂w
∂y
2
= f (x) + g(y).
Complete integral: w =
f (x) + C1dx
g2(y) – C1dy + C2 The signs before each of the integrals can be chosen independently of each other
6.
∂w
∂x
2
+
∂w
∂y
2
= f (x2+ y2 ).
Hamilton’s equation for the plane motion of a point mass under the action of a central force Complete integral:
w = C1arctan x
y + C2 1
2
zf (z) – C12 dz
z , z = x2+ y2
7.
∂w
∂x
2
+
∂w
∂y
2
= f (w).
Complete integral in implicit form:
dw
f (w) =
(x + C1)2+ (y + C2)2
8.
∂w
∂x
2
+ 1
x2
∂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 = C1y
!
f (x) – C
2 1
x2 dx + C2.
9.
∂w
∂x
2
+ f (x)
∂w
∂y
2
= g(x).
Complete integral: w = C1y + C2+
g (x) – C12f (x) dx.
Trang 6
∂w
∂x
2
+ f (y)
∂w
∂y
2
= g(y).
Complete integral: w = C1x + C2+
!g (y) – C2
1
f (y) dy.
11.
∂w
∂x
2
+ f (w)
∂w
∂y
2
= g(w).
Complete integral in implicit form:
!C2
1 + C22f (w)
g (w) dw = C1x + C2y + C3. One of the constants C1or C2can be set equal to 1
12. f1(x)
∂w
∂x
2
+ f2(y)
∂w
∂y
2
= g1(x) + g2(y).
A separable equation This equation is encountered in differential geometry in studying
geodesic lines of Liouville surfaces Complete integral:
w=
!
g1(x) + C1
f1(x) dx
!
g2(y) – C1
f2(y) dy + C2.
The signs before each of the integrals can be chosen independently of each other
T7.3.3 Equations with Arbitrary Nonlinearities in Derivatives
∂x + f
∂w
∂y
= 0.
This equation is encountered in optimal control and differential games
1◦ Complete integral: w = C1y – f (C1)x + C2.
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
∂x + f
∂w
∂y
= g(x).
Complete integral: w = C1y – f (C1)x +
g (x) dx + C2
Trang 71262 FIRST-ORDERPARTIALDIFFERENTIALEQUATIONS
∂x + f
∂w
∂y
= g(x)y + h(x).
Complete integral:
w = ϕ(x)y +
h (x) – f ϕ (x)
dx + C1, where ϕ(x) =
g (x) dx + C2
∂x + f
∂w
∂y
= g(x)w + h(x).
Complete integral:
w = (C1y + C2)ϕ(x) + ϕ(x)
h (x) – f (C1ϕ (x)) dx
ϕ (x) , where ϕ(x) = exp
g (x) dx
∂x – F
x, ∂w
∂y
= 0.
Complete integral: w =
F (x, C1) dx + C1y + C2
∂x + F
x, ∂w
∂y
= aw.
Complete integral: w = e ax (C1y + C2) – e ax
e–ax F (x, C
1e ax ) dx.
∂x + F
x, ∂w
∂y
= g(x)w.
Complete integral:
w = ϕ(x)(C1y + C2) – ϕ(x)
F x , C1ϕ (x) dx
ϕ (x) , where ϕ(x) = exp
g (x) dx
8. F
∂w
∂x, ∂w
∂y
= 0.
Complete integral:
w = C1x + C2y + C3,
where C1and C3are arbitrary constants and the constant C2is related to C1by F (C1, C2) =0
9. w = x ∂w
∂x + y ∂w
∂y + F
∂w
∂x, ∂w
∂y
.
Clairaut’s equation Complete integral: w = C1x + C2y + F (C1, C2)
... (x2+ y2 ).Hamilton’s equation for the plane motion of a point mass under the action of a central force Complete integral:
w = C1arctan...
g2(y) – C1dy + C2 The signs before each of the integrals can be chosen independently of each other
6.
∂w
∂x...
= 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