The functionΦ can also be dependent on the second and higher derivatives with respect to t.. The functionΦ can also be dependent on the second and higher derivatives with respect to t...
Trang 11242 SYSTEMS OFORDINARYDIFFERENTIALEQUATIONS
5. x
tt + a(t)x = x–3f (y/x), y tt + a(t)y = y–3g(y/x).
Generalized Ermakov system.
1◦ First integral:
1
2(xy t – yx t 2+
y/x
uf (u) – u–3g (u)
du = C,
where C is an arbitrary constant.
2◦ Suppose ϕ = ϕ(t) is a nontrivial solution of the second-order linear differential equation
ϕ
Then the transformation
τ =
dt
ϕ2(t), u =
x
ϕ (t), v=
y
leads to the autonomous system of equations
u
ττ = u–3f (v/u), v ττ = v–3g (v/u). (3)
3◦ Particular solution of system (3) is
u = A
C2τ2+ C1τ + C0, v = Ak
C2τ2+ C1τ + C0, A=
f (k)
C0C2– 14C2
1
1 4
,
where C0, C1, and C2are arbitrary constants, and k is a root of the algebraic (transcendental)
equation
k4f (k) = g(k).
6. x
tt = f (y t /x t), y
tt = g(y t /x t).
1◦ The transformation
leads to a system of the first-order equations
u
t = f (w/u), w t = g(w/u). (2)
Eliminating t yields a homogeneous first-order equation, whose solution is given by
f (ξ) dξ
g (ξ) – ξf (ξ) = ln|u|+ C, ξ= w
where C is an arbitrary constant On solving (3) for w, one obtains w = w(u, C) On substituting this expression into the first equation of (2), one can find u = u(t) and then
w = w(t) Finally, one can determine x = x(t) and y = y(t) from (1) by simple integration.
Trang 22◦ The Suslov problem The problem of a point particle sliding down an inclined rough
plane is described by the equations
x
t
(x t2+ (y t 2
, y
tt= –
ky t
(x t 2+ (y t2
,
which correspond to a special case of the system in question with
f (z) =1– √ k
1+ z2, g (z) = –
kz
√
1+ z2.
The solution of the corresponding Cauchy problem under the initial conditions
x(0) = y(0) = x t 0) =0, y
t 0) =1
leads, for the case k =1, to the following dependences x(t) and y(t) written in parametric
form:
x= –161 + 161 ξ4– 1
4ln ξ, y= 23 – 12ξ– 16ξ3, t= 14 – 14ξ2– 12ln ξ (0 ≤ξ≤ 1)
7. x
tt = xΦ(x, y, t, x t , y
t), y
tt = yΦ(x, y, t, x t , y
t).
1◦ First integral:
xy
t – yx t = C,
where C is an arbitrary constant.
Remark The functionΦ can also be dependent on the second and higher derivatives with respect to t.
2◦ Particular solution: y = C1x , where C1is an arbitrary constant and the function x = x(t)
is determined by the ordinary differential equation
x
tt = x Φ(x, C1x , t, x t , C1x
t).
8. x
tt + x–3f (y/x) = xΦ(x, y, t, x t , y
t), y
tt + y–3g(y/x) = yΦ(x, y, t, x t , y
t).
First integral:
1
2(xy t – yx t 2+
y/x
u–3g (u) – uf (u)
du = C, where C is an arbitrary constant.
Remark The functionΦ can also be dependent on the second and higher derivatives with respect to t.
9. x
tt = F (t, tx t – x, ty t – y), y tt = G(t, tx t – x, ty t – y).
1◦ The transformation
leads to a system of first-order equations
u
t = tF (t, u, v), v t = tG(t, u, v). (2)
2◦ Suppose a solution of system (2) has been found in the form
u = u(t, C1, C2), v = v(t, C1, C2), (3)
where C1 and C2 are arbitrary constants Then, substituting (3) into (1) and integrating, one obtains a solution of the original system,
x = C3t + t
u (t, C1, C2)
t2 dt, y = C4t + t
v (t, C1, C2)
t2 dt.
3◦ If the functions F and G are independent of t, then, on eliminating t from system (2),
one arrives at a first-order equation
g (u, v)u v = F (u, v).
Trang 31244 SYSTEMS OFORDINARYDIFFERENTIALEQUATIONS
T6.4 Nonlinear Systems of Three or More Equations
1. ax
t = (b – c)yz, by t = (c – a)zx, cz t = (a – b)xy.
First integrals:
ax2+ by2+ cz2= C
1,
a2x2+ b2y2+ c2z2= C
2,
where C1and C2 are arbitrary constants On solving the first integrals for y and z and on
substituting the resulting expressions into the first equation of the system, one arrives at a separable first-order equation
2. ax
t = (b – c)yzF (x, y, z, t),
by
t = (c – a)zxF (x, y, z, t), cz t = (a – b)xyF (x, y, z, t).
First integrals:
ax2+ by2+ cz2= C
1,
a2x2+ b2y2+ c2z2= C
2,
where C1and C2 are arbitrary constants On solving the first integrals for y and z and on
substituting the resulting expressions into the first equation of the system, one arrives at a
separable first-order equation; if F is independent of t, this equation will be separable.
3. x
t = cF2– bF3 , y
t = aF3– cF1 , z
t = bF1– aF2 , where F n = F n (x, y, z).
First integral:
ax + by + cz = C1,
where C1 is an arbitrary constant On eliminating t and z from the first two equations of
the system (using the above first integral), one arrives at the first-order equation
dy
dx = aF3(x, y, z) – cF1(x, y, z)
cF2(x, y, z) – bF3(x, y, z), where z=
1
c (C1– ax – by).
4. x
t = czF2– byF3 , y
t = axF3– czF1 , z
t = byF1– axF2
Here, F n = F n (x, y, z) are arbitrary functions (n =1, 2, 3)
First integral:
ax2+ by2+ cz2= C
1,
where C1 is an arbitrary constant On eliminating t and z from the first two equations of
the system (using the above first integral), one arrives at the first-order equation
dy
dx = axF3(x, y, z) – czF1(x, y, z)
czF2(x, y, z) – byF3(x, y, z), where z=
1
c (C1– ax2– by2)
5. x
t = x(cF2– bF3 ), y
t = y(aF3– cF1 ), z
t = z(bF1– aF2 ).
Here, F n = F n (x, y, z) are arbitrary functions (n =1, 2, 3)
First integral:
|x|a|y|b|z|c = C1,
where C1 is an arbitrary constant On eliminating t and z from the first two equations of
the system (using the above first integral), one may obtain a first-order equation
Trang 46. x
t = h(z)F2– g(y)F3 , y
t = f (x)F3– h(z)F1 , z
t = g(y)F1– f (x)F2
Here, F n = F n (x, y, z) are arbitrary functions (n =1, 2, 3)
First integral:
f (x) dx +
g (y) dy +
h (z) dz = C1,
where C1 is an arbitrary constant On eliminating t and z from the first two equations of
the system (using the above first integral), one may obtain a first-order equation
7. x
tt=
∂F
∂x, y
tt=
∂F
∂y, z
tt=
∂F
∂z , where F = F (r), r =
x2+ y2+ z2
Equations of motion of a point particle under gravity.
The system can be rewritten as a single vector equation:
rtt = grad F or rtt = F (r)
r r,
where r = (x, y, z).
1◦ First integrals:
(r t 2 =2F(r) + C1 (law of conservation of energy),
[r×r t] = C (law of conservation of areas),
(r⋅C) =0 (all trajectories are plane curves)
2◦ Solution:
r = a r cos ϕ + b r sin ϕ.
Here, the constant vectors a and b must satisfy the conditions
|a|=|b|=1, (a⋅b) =0,
and the functions r = r(t) and ϕ = ϕ(t) are given by
t=
r dr
2r2F (r) + C
1r2– C2 3
+ C2, ϕ = C3
dr
r 2r2F (r) + C
1r2– C2 3 , C3 =|C|
8. x
tt = xF , y tt = yF , z tt = zF , where F = F (x, y, z, t, x t , y t , z t ).
First integrals (laws of conservation of areas):
zy
t – yz t = C1,
xz
t – zx t = C2,
yx
t – xy t = C3,
where C1, C2, and C3are arbitrary constants
Corollary of the conservation laws:
C1x + C2y + C3z=0 This implies that all integral curves are plane ones
Remark The functionΦ can also be dependent on the second and higher derivatives with respect to t.
Trang 51246 SYSTEMS OFORDINARYDIFFERENTIALEQUATIONS
9. x
tt = F1 , y
tt = F2 , z
tt = F3 , where F n = F n (t, tx t – x, ty t – y, tz t – z).
1◦ The transformation
u = tx t – x, v = ty t – y, w = tz t – z (1) leads to the system of first-order equations
u
t = tF1(t, u, v, w), v t = tF2(t, u, v, w), w t = tF3(t, u, v, w). (2)
2◦ Suppose a solution of system (2) has been found in the form
u (t) = u(t, C1, C2, C3), v (t) = v(t, C1, C2, C3), w (t) = w(t, C1, C2, C3), (3)
where C1, C2, and C3are arbitrary constants Then, substituting (3) into (1) and integrating, one obtains a solution of the original system:
x = C4t + t
u (t)
t2 dt, y = C5t + t
v (t)
t2 dt, z = C6t + t
w (t)
t2 dt,
where C4, C5, and C6are arbitrary constants
References for Chapter T6
Kamke, E., Differentialgleichungen: L¨osungsmethoden und L ¨osungen, I, Gew¨ohnliche Differentialgleichungen,
B G Teubner, Leipzig, 1977.
Polyanin, A D., Systems of Ordinary Differential Equations, From Website EqWorld—The World of
Mathe-matical Equations, http://eqworld.ipmnet.ru/en/solutions/sysode.htm.
Polyanin, A D and Zaitsev, V F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd
Edition, Chapman & Hall/CRC Press, Boca Raton, 2003.
Trang 6First-Order Partial
Differential Equations
T7.1 Linear Equations
In equations T7.1.1.1–T7.1.1.11, the general solution is expressed in terms of the prin-cipal integral Ξ as w = Φ(Ξ), where Φ(Ξ) is an arbitrary function.
∂x +
f (x)y + g(x) ∂w
∂y = 0.
Principal integral: Ξ = e–F y–
e–F g (x) dx, where F =
f (x) dx.
∂x +
f (x)y + g(x)y k ∂w
∂y = 0.
Principal integral: Ξ = e–F y1 –k– (1– k)
e–F g (x) dx, where F = (1– k)
f (x) dx.
∂x +
f (x)e λy + g(x) ∂w
∂y = 0.
Principal integral: Ξ = e–λy E + λ
f (x)E dx, where E = exp
λ
g (x) dx
4. f (x) ∂w
∂x + g(y) ∂w
∂y = 0.
Principal integral: Ξ =
dx
f (x) –
dy
g (y).
5.
f (y) + amx n y m–1 ∂w
∂x –
g(x) + anx n–1 y m ∂w
∂y = 0.
Principal integral: Ξ =
f (y) dy +
g (x) dx + ax n y m.
6.
e αx f (y) + cβ ∂w
∂x –
e βy g(x) + cα ∂w
∂y = 0.
Principal integral: Ξ =
e–βy f (y) dy +
e–αx g (x) dx – ce–αx–βy.
1247
Trang 71248 FIRST-ORDERPARTIALDIFFERENTIALEQUATIONS
∂x + f (ax + by + c) ∂w
Principal integral: Ξ =
dv
a + bf (v) – x, where v = ax + by + c.
∂x + f
y x
∂w
∂y = 0.
Principal integral: Ξ =
dv
f (v) – v – ln|x|, where v = y
x
9. x ∂w
∂x + yf (x n y m)∂w
∂y = 0.
Principal integral: Ξ =
dv
v
mf (v) + n – ln|x|, where v = x n y m.
10. ∂w
∂x + yf (e αx y m)∂w
∂y = 0.
Principal integral: Ξ =
dv
v
α + mf (v) – x, where v = e αx y m.
11. x ∂w
∂x + f (x n e αy)∂w
∂y = 0.
Principal integral: Ξ =
dv
v
n + αf (v) – ln|x|, where v = x n e αy.
In the solutions of equations T7.1.2.1–T7.1.2.12, Φ(z) is an arbitrary composite function whose argument z can depend on both x and y.
1. a ∂w
∂y = f (x).
General solution: w = 1
a
f (x) dx + Φ(bx – ay).
∂y = f (x)y k.
General solution: w =
x
x0
(y –ax+at) k f (t) dt+ Φ(y–ax), where x0can be taken arbitrarily.
∂y = f (x)e λy.
General solution: w = e λ(y–ax)
f (x)e aλx dx+Φ(y – ax).
... C1and C2 are arbitrary constants On solving the first integrals for y and z and onsubstituting the resulting expressions into the first equation of the system,... C1and C2 are arbitrary constants On solving the first integrals for y and z and on
substituting the resulting expressions into the first equation of the system,... http://eqworld.ipmnet.ru/en/solutions/sysode.htm.
Polyanin, A D and Zaitsev, V F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd
Edition,