2 In order to find the general solution of this system, it suffices to know its any particular solution see system T6.1.1.7.. The system concerned can be reduced to a Riccati equation se
Trang 1where C1 and C2 are arbitrary constants On substituting (2) into (1) and integrating, one arrives at the general solution of the original system in the form
x = C3t + t
u (t)
t2 dt, y = C4t + t
v (t)
t2 dt,
where C3and C4are arbitrary constants
6. x
tt = f (t)(a1x + b1y), y
tt = f (t)(a2x + b2y).
Let k1and k2be roots of the quadratic equation
k2– (a
1+ b2)k + a1b2– a2b1=0
Then, on multiplying the equations of the system by appropriate constants and on adding them together, one can rewrite the system in the form of two independent equations:
z
1 = k1f (t)z1, z1= a2x + (k1– a1)y;
z
2 = k2f (t)z2, z2= a2x + (k2– a1)y.
Here, a prime stands for a derivative with respect to t.
7. x
tt = f (t)(a1x
t + b1y
t), y
tt = f (t)(a2x
t + b2y
t).
Let k1and k2be roots of the quadratic equation
k2– (a
1+ b2)k + a1b2– a2b1=0
Then, on multiplying the equations of the system by appropriate constants and on adding them together, one can reduce the system to two independent equations:
z
1 = k1f (t)z1, z1= a2x + (k1– a1)y;
z
2 = k2f (t)z2, z2= a2x + (k2– a1)y.
Integrating these equations and returning to the original variables, one arrives at a linear
algebraic system for the unknowns x and y:
a2x + (k1– a1)y = C1
exp
k1F (t)
dt + C2,
a2x + (k2– a1)y = C3
exp
k2F (t)
dt + C4,
where C1, , C4are arbitrary constants and F (t) =
f (t) dt.
8. x
tt = af (t)(ty t – y), y tt = bf (t)(tx t – x).
The transformation
leads to a system of first-order equations:
u
t = atf (t)v, v t = btf (t)u.
Trang 21236 SYSTEMS OFORDINARYDIFFERENTIALEQUATIONS
The general solution of this system is expressed as
if ab >0 ,
⎧
⎪
⎪
u(t) = C1aexp√
ab
tf(t) dt
+ C2aexp
–√ ab
tf(t) dt
,
v(t) = C1√
ab exp√
ab
tf(t) dt
– C2√
abexp
–√ ab
tf (t) dt
;
if ab <0 ,
⎧
⎪
⎪
u (t) = C1acos
|ab|
tf (t) dt
+ C2asin
|ab|
tf (t) dt
,
v(t) = –C1
|ab| sin
|ab| tf(t) dt
+ C2
|ab| cos
|ab| tf (t) dt
, (2)
where C1 and C2 are arbitrary constants On substituting (2) into (1) and integrating, one obtains the general solution of the original system
x = C3t + t
u (t)
t2 dt, y = C4t + t
v (t)
t2 dt,
where C3and C4are arbitrary constants
9. t2x
tt + a1tx
t + b1ty
t + c1x + d1y= 0, t2y
tt + a2tx
t + b2ty
t + c2x + d2y = 0.
Linear system homogeneous in the independent variable (an Euler type system).
1◦ The general solution is determined by a linear combination of linearly independent
particular solutions that are sought by the method of undetermined coefficients in the form
of power-law functions
x = A| t|k, y = B| t|k.
On substituting these expressions into the system and on collecting the coefficients of the
unknowns A and B, one obtains
[k2+ (a1–1)k + c1]A + (b1k + d1)B =0,
(a2k + c2)A + [k2+ (b2–1)k + d2]B =0
For a nontrivial solution to exist, the determinant of this system must vanish This require-ment results in the characteristic equation
[k2+ (a1–1)k + c1][k2+ (b2–1)k + d2] – (b1k + d1)(a2k + c2) =0,
which is used to determine k If the roots of this equation, k1, , k4, are all distinct, then the general solution of the system of differential equations in question has the form
x = –C1(b1k1+ d1)|t|k1 – C2(b1k2+ d1)|t|k2– C3(b1k1+ d1)|t|k3 – C4(b1k4+ d1)|t|k4,
y = C1[k12+ (a1–1)k1+ c1]|t|k1 + C2[k22+ (a1–1)k2+ c1]|t|k2
+ C3[k23+ (a1–1)k3+ c1]|t|k3+ C4[k42+ (a1–1)k4+ c1]|t|k4,
where C1, , C4are arbitrary constants
2◦ The substitution t = σe τ (σ ≠ 0) leads to a system of constant-coefficient linear differential equations:
x
ττ + (a1–1)x
τ + b1y
τ + c1x + d1y=0,
y
ττ + a2x
τ + (b2–1)y
τ + c2x + d2y=0
Trang 310. (αt2+ βt + γ)2x
tt = ax + by, (αt2+ βt + γ)2y
tt = cx + dy.
The transformation
τ =
dt
αt2+ βt + γ, u=
x
|αt2+ βt + γ|, v=
y
|αt2+ βt + γ|
leads to a constant-coefficient linear system of equations of the form T6.1.2.1:
u
ττ = (a – αγ + 14β2)u + bv,
v
ττ = cu + (d – αγ + 14β2)v.
11. x
tt = f (t)(tx t – x) + g(t)(ty t – y), y tt = h(t)(tx t – x) + p(t)(ty t – y).
The transformation
leads to a linear system of first-order equations
u
t = tf (t)u + tg(t)v, v t = th(t)u + tp(t)v. (2)
In order to find the general solution of this system, it suffices to know its any particular solution (see system T6.1.1.7)
For solutions of some systems of the form (2), see systems T6.1.1.3–T6.1.1.6
If all functions in (2) are proportional, that is,
f (t) = aϕ(t), g (t) = bϕ(t), h (t) = cϕ(t), p (t) = dϕ(t), then the introduction of the new independent variable τ =
tϕ (t) dt leads to a
constant-coefficient system of the form T6.1.1.1
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 C1and C2are arbitrary constants Then, on 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,
where C3and C4are arbitrary constants
T6.2 Linear Systems of Three and More Equations
1. x
t = ax, y t = bx + cy, z t = dx + ky + pz.
Solution:
x = C1e at,
y = bC1
a – c e
at + C
2e ct,
z= C1
a – p
d+ bk
a – c
e at+ kC2
c – p e
ct + C3e pt,
where C1, C2, and C3are arbitrary constants
Trang 41238 SYSTEMS OFORDINARYDIFFERENTIALEQUATIONS
2. x
t = cy – bz, y t = az – cx, z t = bx – ay.
1◦ First integrals:
where A and B are arbitrary constants It follows that the integral curves are circles formed
by the intersection of planes (1) and spheres (2)
2◦ Solution:
x = aC0+ kC1cos(kt) + (cC2– bC3) sin(kt),
y = bC0+ kC2cos(kt) + (aC3– cC1) sin(kt),
z = cC0+ kC3cos(kt) + (bC1– aC2) sin(kt), where k = √
a2+ b2+ c2and the three of four constants of integration C0, , C3are related
by the constraint
aC1+ bC2+ cC3=0
3. ax
t = bc(y – z), by t = ac(z – x), cz t = ab(x – y).
1◦ First integral:
a2x + b2y + c2z = A,
where A is an arbitrary constant It follows that the integral curves are plane ones.
2◦ Solution:
x = C0+ kC1cos(kt) + a–1bc (C2– C3) sin(kt),
y = C0+ kC2cos(kt) + ab–1c (C3– C1) sin(kt),
z = C0+ kC3cos(kt) + abc–1(C1– C2) sin(kt), where k = √
a2+ b2+ c2and the three of four constants of integration C0, , C3are related
by the constraint
a2C
1+ b2C2+ c2C3=0
4. x
t = (a1f + g)x + a2f y + a3f z,
y
t = b1f x + (b2f + g)y + b3f z, z
t = c1f x + c2f y + (c3f + g)z.
Here, f = f (t) and g = g(t).
The transformation
x= exp
g (t) dt
u, y= exp
g (t) dt
v, z= exp
g (t) dt
w, τ =
f (t) dt
leads to the system of constant coefficient linear differential equations
u
τ = a1u + a2v + a3w, v
τ = b1u + b2v + b3w, w
τ = c1u + c2v + c3w
5. x
t = h(t)y – g(t)z, y t = f (t)z – h(t)x, z t = g(t)x – f (t)y.
1◦ First integral:
x2+ y2+ z2= C2,
where C is an arbitrary constant.
2◦ The system concerned can be reduced to a Riccati equation (see Kamke, 1977).
Trang 56. x
k = a k1 x1+ a k2 x2 +· · · + a kn x n; k = 1, 2, , n.
System of n constant-coefficient first-order linear homogeneous differential equations.
The general solution of a homogeneous system of differential equations is determined
by a linear combination of linearly independent particular solutions, which are sought by the method of undetermined coefficients in the form of exponential functions,
x k = A k e λt; k=1,2, , n
On substituting these expressions into the system and on collecting the coefficients of the
unknowns A k, one obtains a linear homogeneous system of algebraic equations:
a k1A1+ a k2A2+· · · + (a kk – λ)A k+· · · + a kn A n=0; k=1,2, , n
For a nontrivial solution to exist, the determinant of this system must vanish This
require-ment results in a characteristic equation that serves to determine λ.
T6.3 Nonlinear Systems of Two Equations
T6.3.1 Systems of First-Order Equations
1. x
t = x n F (x, y), y t = g(y)F (x, y).
Solution:
x = ϕ(y),
dy
g (y)F (ϕ(y), y) = t + C2,
where
ϕ (y) =
⎧
⎪
⎨
⎪
⎩
*
C1+ (1– n)
dy
g (y)
+ 1 1–n if n≠ 1,
C1exp
* dy
g (y)
+
if n =1,
C1and C2are arbitrary constants.
2. x
t = e λx F (x, y), y t = g(y)F (x, y).
Solution:
x = ϕ(y),
dy
g (y)F (ϕ(y), y) = t + C2,
where
ϕ (y) =
⎧
⎪
⎪
–1
λln
*
C1– λ
dy
g (y)
+
if λ≠ 0,
C1+
dy
C1and C2are arbitrary constants.
Trang 61240 SYSTEMS OFORDINARYDIFFERENTIALEQUATIONS
3. x
t = F (x, y), y t = G(x, y).
Autonomous system of general form.
Suppose
y = y(x, C1),
where C1is an arbitrary constant, is the general solution of the first-order equation
F (x, y)y x = G(x, y).
Then the general solution of the system in question results in the following dependence for
dx
F (x, y(x, C1)) = t + C2.
4. x
t = f1(x)g1(y)Φ(x, y, t), y
t = f2(x)g2(y)Φ(x, y, t).
f2(x)
f1(x) dx–
g1(y)
where C is an arbitrary constant.
On solving (∗) for x (or y) and on substituting the resulting expression into one of the
equations of the system concerned, one arrives at a first-order equation for y (or x).
5. x = tx t + F (x t , y t), y = ty t + G(x t , y t).
Clairaut system.
The following are solutions of the system:
(i) straight lines
x = C1t + F (C1, C2), y = C2t + G(C1, C2),
where C1and C2are arbitrary constants;
(ii) envelopes of these lines;
(iii) continuously differentiable curves that are formed by segments of curves (i) and (ii)
T6.3.2 Systems of Second-Order Equations
1. x
tt = xf (ax – by) + g(ax – by), y tt = yf (ax – by) + h(ax – by).
Let us multiply the first equation by a and the second one by –b and add them together to
obtain the autonomous equation
z
tt = zf (z) + ag(z) – bh(z), z = ax – by. (1)
We will consider this equation in conjunction with the first equation of the system,
x
Equation (1) can be treated separately; its general solution can be written out in implicit
form (see Polyanin and Zaitsev, 2003) The function x = x(t) can be determined by solving the linear equation (2), and the function y = y(t) is found as y = (ax – z)/b.
Trang 72. x
tt = xf (y/x), y tt = yg(y/x).
A periodic particular solution:
x = C1sin(kt) + C2cos(kt), k=
–f (λ),
y = λ[C1sin(kt) + C2cos(kt)], where C1 and C2 are arbitrary constants and λ is a root of the transcendental (algebraic)
equation
2◦ Particular solution:
x = C1exp(kt) + C2exp(–kt), k=
f (λ),
y = λ[C1exp(kt) + C2exp(–kt)], where C1 and C2 are arbitrary constants and λ is a root of the transcendental (algebraic)
equation (1)
3. x
tt = kxr–3 , y
tt = kyr–3 , where r=
x2+ y2
Equation of motion of a point mass in the xy-plane under gravity.
Passing to polar coordinates by the formulas
x = r cos ϕ, y = r sin ϕ, r = r(t), ϕ = ϕ(t),
one may obtain the first integrals
r2ϕ
t = C1, (r t2+ r2(ϕ t 2= –2kr–1+ C
where C1and C2are arbitrary constants Assuming that C1≠ 0and integrating further, one finds that
r [C cos(ϕ – ϕ0) – k] = C12, C2= C2
1C2+ k2.
This is an equation of a conic section The dependence ϕ(t) may be found from the first
equation in (1)
4. x
tt = xf (r), y tt = yf (r), where r=
x2+ y2
Equation of motion of a point mass in the xy-plane under a central force.
Passing to polar coordinates by the formulas
x = r cos ϕ, y = r sin ϕ, r = r(t), ϕ = ϕ(t),
one may obtain the first integrals
r2ϕ
t = C1, (r t 2+ r2(ϕ t2 =2 rf (r) dr + C2,
where C1and C2are arbitrary constants Integrating further, one finds that
t + C3=
r dr
2r2F (r) + r2C2– C2
1
, ϕ = C1
dt
where C3and C4are arbitrary constants and
F (r) =
rf (r) dr.
It is assumed in the second relation in (∗) that the dependence r = r(t) is obtained by solving
the first equation in (∗) for r(t).
... order to find the general solution of this system, it suffices to know its any particular solution (see system T6.1.1.7)For solutions of some systems of the form (2), see systems T6.1.1.3–T6.1.1.6... determined by a linear combination of linearly independent
particular solutions that are sought by the method of undetermined coefficients in the form
of power-law functions
x... determined
by a linear combination of linearly independent particular solutions, which are sought by the method of undetermined coefficients in the form of exponential functions,
x