Let us note some special properties of the Sturm–Liouville problem that is the mixed boundary value problem for equation 12.2.5.1 with the boundary conditions y x=0 at x = x1, y=0 at x
Trang 1486 ORDINARYDIFFERENTIALEQUATIONS
2◦ Suppose p(x) = s(x) = 1 and the function q = q(x) has a continuous derivative The following asymptotic relations hold for eigenvalues λ n and eigenfunctions y n (x) as
n → ∞:
λ n= π (n –1)
x2– x1 +
1
π (n –1)Q (x1, x2) + O
1
n2
,
y n (x) = cos π (n –1)(x – x1)
x2– x1 +
1
π (n –1)
*
(x1– x)Q(x, x2)
+ (x2– x)Q(x1, x)
+ sin π (n –1)(x – x1)
x2– x1 + O
1
n2
,
where Q(u, v) is given by (12.2.5.8).
12.2.5-5 Problems with boundary conditions of the third kind
We consider the third boundary value problem for equation (12.2.5.1) subject to
condi-tion (12.2.5.2) with α1= α2 =1 We assume that p(x) = s(x) =1 and the function q = q(x)
has a continuous derivative
The following asymptotic formulas hold for eigenvalues λ n and eigenfunctions y n (x)
as n → ∞:
λ n= π (n –1)
x2– x1 +
1
π (n –1)
Q (x1, x2) – β1+ β2
+ O 1
n2
,
y n (x) = cos π (n –1)(x – x1)
x2– x1 +
1
π (n –1)
(x1– x)
Q (x, x2) + β2
+ (x2– x)
Q (x1, x) – β14
sinπ (n –1)(x – x1)
x2– x1 + O
1
n2
,
where Q(u, v) is defined by (12.2.5.8).
12.2.5-6 Problems with mixed boundary conditions
Let us note some special properties of the Sturm–Liouville problem that is the mixed boundary value problem for equation (12.2.5.1) with the boundary conditions
y
x=0 at x = x1, y=0 at x = x2
1◦ If q≥ 0, the upper estimate (12.2.5.6) is valid for the least eigenvalue, with z = z(x) being any twice-differentiable function that satisfies the conditions z x (x1) =0 and z(x2) =0
The equality in (12.2.5.6) is attained if z = y1(x), where y1(x) is the eigenfunction corresponding to the eigenvalue λ1
2◦ Suppose p(x) = s(x) = 1 and the function q = q(x) has a continuous derivative The following asymptotic relations hold for eigenvalues λ n and eigenfunctions y n (x) as
n → ∞:
λ n= π(2n–1)
2(x2– x1) +
2
π(2n–1)Q (x1, x2) + O
1
n2
,
y n (x) = cos π(2n–1)(x – x1)
2(x2– x1) +
2
π(2n–1)
*
(x1– x)Q(x, x2)
+ (x2– x)Q(x1, x)
+ sin π(2n–1)(x – x1)
2(x2– x1) + O
1
n2
,
where Q(u, v) is defined by (12.2.5.8).
Trang 212.2.6 Theorems on Estimates and Zeros of Solutions
12.2.6-1 Theorems on estimates of solutions
Let f n (x) and g n (x) (n =1, 2) be continuous functions on the interval a≤x≤band let the following inequalities hold:
0 ≤f1(x)≤f2(x), 0 ≤g1(x)≤g2(x).
If y n = y n (x) are some solutions to the linear equations
y
n = f n (x)y n + g n (x) (n =1,2)
and y1(a) ≤y2(a) and y 1(a) ≤y
2(a), then y1(x)≤y2(x) and y 1(x)≤y
2(x) on each interval
a≤x≤a1, where y2(x) >0
12.2.6-2 Sturm comparison theorem on zeros of solutions
Consider the equation
[f (x)y ] + g(x)y =0 (a≤x≤b), (12.2.6.1)
where the function f (x) is positive and continuously differentiable, and the function g(x)
is continuous
THEOREM(COMPARISON, STURM) Let y n = y n (x) be nonzero solutions of the linear equations
[f n (x)y n] + g n (x)y n=0 (n =1, 2)
and let the inequalities f1(x) ≥f2(x) > 0 and g1(x) ≤ g2(x) hold Then the function y2 has at least one zero lying between any two adjacent zeros, x1 and x2, of the function y1 (it is assumed that the identities f1 ≡ f2 and g1 ≡ g2 are not satisfied on any interval simultaneously)
COROLLARY1 If g(x) ≤ 0or there exists a constant k1such that
f (x)≥k1>0, g (x) < k1
π
b – a
2 ,
then every nontrivial solution to equation (12.2.6.1) has no more than one zero on the
interval [a, b].
COROLLARY2 If there exists a constant k2such that
0< f (x)≤k2, g (x) > k2
πm
b – a
2 , where m =1, 2, , then every nontrivial solution to equation (12.2.6.1) has at least m zeros on the interval [a, b].
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12.2.6-3 Qualitative behavior of solutions as x → ∞.
Consider the equation
where f (x) is a continuous function for x≥a
1◦ For f (x)≤ 0, every nonzero solution has no more than one zero, and hence y ≠ 0for
sufficiently large x.
If f (x)≤ 0for all x and f (x)0, then y≡ 0is the only solution bounded for all x.
2◦ Suppose f (x) ≥ k2 > 0 Then every nontrivial solution y(x) and its derivative y (x)
have infinitely many zeros, with the distance between any adjacent zeros remaining finite
If f (x) → k2>0for x → ∞ and f ≥ 0, then the solutions of the equation for large x behave similarly to those of the equation y + k2y=0
3◦ Let f (x) → –∞ for|x| → ∞ Then every nonzero solution has only finitely many
zeros, and|y /y|→ ∞ as|x|→ ∞ There are two linearly independent solutions, y1and y2,
such that y1→0, y 1→0, y2→ ∞, and y
2 → –∞ as x → –∞, and there are two linearly
independent solutions, ¯y1 and ¯y2, such that ¯y1 → 0, ¯y
1 → 0, ¯y2 → ∞, and ¯y
2 → ∞ as
x → ∞.
4◦ If the function f in equation (12.2.6.2) is continuous, monotonic, and positive, then the
amplitude of each solution decreases (resp., increases) as f increases (resp., decreases).
12.3 Second-Order Nonlinear Differential Equations
12.3.1 Form of the General Solution Cauchy Problem
12.3.1-1 Equations solved for the derivative General solution
A second-order ordinary differential equation solved for the highest derivative has the form
y
xx = f (x, y, y x ). (12.3.1.1)
The general solution of this equation depends on two arbitrary constants, C1 and C2 In
some cases, the general solution can be written in explicit form, y = ϕ(x, C1, C2), but more often implicit or parametric forms of the general solution are encountered
12.3.1-2 Cauchy problem The existence and uniqueness theorem
Cauchy problem: Find a solution of equation (12.3.1.1) satisfying the initial conditions
y (x0) = y0, y
x (x0) = y1. (12.3.1.2)
(At a point x = x0, the value of the unknown function, y0, and its derivative, y1, are prescribed.)
EXISTENCE AND UNIQUENESS THEOREM Let f (x, y, z) be a continuous function in all its arguments in a neighborhood of a point (x0, y0, y1) and let f have bounded par-tial derivatives f y and f z in this neighborhood, or the Lipschitz condition is satisfied:
|f (x, y, z) – f (x, ¯y, ¯z)| ≤ A |y– ¯y|+|z– ¯z| , where A is some positive number Then
a solution of equation (12.3.1.1) satisfying the initial conditions (12.3.1.2) exists and is unique
Trang 412.3.2 Equations Admitting Reduction of Order
12.3.2-1 Equations not containing y explicitly.
In the general case, an equation that does not contain y implicitly has the form
F (x, y x , y xx) =0 (12.3.2.1) Such equations remain unchanged under an arbitrary translation of the dependent variable:
y → y + const The substitution y
x = z(x), y xx = z x (x) brings (12.3.2.1) to a first-order
equation: F (x, z, z x ) =0
12.3.2-2 Equations not containing x explicitly (autonomous equations).
In the general case, an equation that does not contain x implicitly has the form
F (y, y x , y xx ) =0 (12.3.2.2) Such equations remain unchanged under an arbitrary translation of the independent
vari-able: x → x + const Using the substitution y
x = w(y), where y plays the role of the
independent variable, and taking into account the relations y xx = w x = w y y
x = w y w, one
can reduce (12.3.2.2) to a first-order equation: F (y, w, ww y) =0
Example 1 Consider the autonomous equation
y xx = f (y),
which often arises in the theory of heat and mass transfer and combustion The change of variable y x =
w (y) leads to a separable first-order equation: ww y = f (y) Integrating yields w2 = 2F(w) + C1, where
F (w) =7
f (w) dw Solving for w and returning to the original variable, we obtain the separable equation
y x= √
2F(w) + C1 Its general solution is expressed as
dy
√
2F(w) + C1 = x + C2, where F (w) =
f (w) dw.
Remark. The equation y xx = f (y + ax2+ bx + c) is reduced by the change of variable u = y + ax2+ bx + c
to an autonomous equation, u xx = f (u) +2a
12.3.2-3 Equations of the form F (ax + by, y x , y xx ) =0
Such equations are invariant under simultaneous translations of the independent and
depen-dent variables in accordance with the rule x → x + bc, y → y – ac, where c is an arbitrary
constant
For b =0, see equation (12.3.2.1) For b≠ 0, the substitution bw = ax + by leads to equation (12.3.2.2): F (bw, w x – a/b, w xx ) =0
12.3.2-4 Equations of the form F (x, xy x – y, y xx) =0
The substitution w(x) = xy x – y leads to a first-order equation: F (x, w, w x /x) =0
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12.3.2-5 Homogeneous equations
1◦ The equations homogeneous in the independent variable remain unchanged under
scaling of the independent variable, x → αx, where α is an arbitrary nonzero number In
the general case, such equations can be written in the form
F (y, xy x , x2y
xx) =0 (12.3.2.3)
The substitution z(y) = xy x leads to a first-order equation: F (y, z, zz y – z) =0
2◦ The equations homogeneous in the dependent variable remain unchanged under scaling
of the variable sought, y → αy, where α is an arbitrary nonzero number In the general
case, such equations can be written in the form
F (x, y x /y , y xx /y) =0 (12.3.2.4)
The substitution z(x) = y x /y leads to a first-order equation: F (x, z, z x + z2) =0
3◦ The equations homogeneous in both variables are invariant under simultaneous scaling
(dilatation) of the independent and dependent variables, x → αx and y → αy, where α is
an arbitrary nonzero number In the general case, such equations can be written in the form
F (y/x, y x , xy xx) =0 (12.3.2.5)
The transformation t = ln|x|, w = y/x leads to an autonomous equation (see Paragraph 12.3.2-2): F (w, w t + w, w tt + w t) =0
Example 2 The homogeneous equation
xy xx – y x = f (y/x)
is reduced by the transformation t = ln|x|, w = y/x to the autonomous form: w tt = f (w) + w For solution
of this equation, see Example 1 in Paragraph 12.3.2-2 (the notation of the right-hand side has to be changed there).
12.3.2-6 Generalized homogeneous equations
1◦ The generalized homogeneous equations remain unchanged under simultaneous scaling
of the independent and dependent variables in accordance with the rule x → αx and
y → α k y , where α is an arbitrary nonzero number and k is some number Such equations
can be written in the form
F (x–k y , x1–k y
x , x2–k y xx) =0 (12.3.2.6)
The transformation t = ln x, w = x–k y leads to an autonomous equation (see Paragraph 12.3.2-2):
F w , w t + kw, w tt+ (2k–1)w t + k(k –1)w
=0
2◦ The most general form of representation of generalized homogeneous equations is as
follows:
F(x n y m , xy
x /y , x2y xx /y) =0 (12.3.2.7)
The transformation z = x n y m , u = xy
x /y brings this equation to the first-order equation
F z , u, z(mu + n)u z – u + u2
=0 Remark. For m≠ 0, equation (12.3.2.7) is equivalent to equation (12.3.2.6) in which k = –n/m To the particular values n =0 and m =0 there correspond equations (12.3.2.3) and (12.3.2.4) homogeneous in
the independent and dependent variables, respectively For n = –m≠ 0 , we have an equation homogeneous in both variables, which is equivalent to equation (12.3.2.5).
Trang 612.3.2-7 Equations invariant under scaling–translation transformations.
1◦ The equations of the form
F (e λx y , e λx y
x , e λx y xx) =0 (12.3.2.8)
remain unchanged under simultaneous translation and scaling of variables, x → x + α
and y → βy, where β = e–αλ and α is an arbitrary number The substitution w = e λx y
brings (12.3.2.8) to an autonomous equation: F (w, w x – λw, w xx–2λw
x + λ2w) =0(see Paragraph 12.3.2-2)
2◦ The equation
F (e λx y n , y
x /y , y xx /y) =0 (12.3.2.9)
is invariant under the simultaneous translation and scaling of variables, x → x + α and
y → βy, where β = e–αλ/n and α is an arbitrary number The transformation z = e λx y n,
w = y x /y brings (12.3.2.9) to a first-order equation: F z , w, z(nw + λ)w z + w2
=0
3◦ The equation
F (x n e λy , xy
x , x2y xx ) =0 (12.3.2.10)
is invariant under the simultaneous scaling and translation of variables, x → αx and
y → y + β, where α = e–βλ/n and β is an arbitrary number The transformation z = x n e λy,
w = xy x brings (12.3.2.10) to a first-order equation: F z , w, z(λw + n)w z – w
=0
Some other second-order nonlinear equations are treated in Section T5.3
12.3.2-8 Exact second-order equations
The second-order equation
F (x, y, y x , y xx) =0 (12.3.2.11)
is said to be exact if it is the total differential of some function, F = ϕ x , where ϕ = ϕ(x, y, y x)
If equation (12.3.2.11) is exact, then we have a first-order equation for y:
ϕ (x, y, y x ) = C, (12.3.2.12)
where C is an arbitrary constant.
If equation (12.3.2.11) is exact, then F (x, y, y x , y xx) must have the form
F (x, y, y x , y xx ) = f (x, y, y x )y xx + g(x, y, y x ) (12.3.2.13)
Here, f and g are expressed in terms of ϕ by the formulas
f (x, y, y x) = ∂ϕ
∂y x, g (x, y, y x) =
∂ϕ
∂x + ∂ϕ
∂y y
x. (12.3.2.14)
By differentiating (12.3.2.14) with respect to x, y, and p = y x , we eliminate the
variable ϕ from the two formulas in (12.3.2.14) As a result, we have the following test relations for f and g:
f xx+2pf xy + p2f yy = g xp + pg yp – g y,
f xp + pf yp+2f y = g pp. (12.3.2.15) Here, the subscripts denote the corresponding partial derivatives
Trang 7492 ORDINARYDIFFERENTIALEQUATIONS
If conditions (12.3.2.15) hold, then equation (12.3.2.11) with F of (12.3.2.13) is exact.
In this case, we can integrate the first equation in (12.3.2.14) with respect to p = y x to
determine ϕ = ϕ(x, y, y x ):
ϕ= f (x, y, p) dp + ψ(x, y), (12.3.2.16)
where ψ(x, y) is an arbitrary function of integration This function is determined by
substituting (12.3.2.16) into the second equation in (12.3.2.14)
Example 3 The left-hand side of the equation
yy xx + (y x )2+ 2axyy
can be represented in the form (12.3.2.13), where f = y and g = p2+ 2axyp+ ay2 It is easy to verify that conditions (12.3.2.15) are satisfied Hence, equation (12.3.2.17) is exact Using (12.3.2.16), we obtain
ϕ = yp + ψ(x, y). (12 3 2 18 )
Substituting this expression into the second equation in (12.3.2.14) and taking into account the relation g =
p2+ 2axyp+ ay2, we find that 2axyp+ ay2 = ψ x + pψ y Since ψ = ψ(x, y), we have 2axy = ψ y and
ay2= ψ x Integrating yields ψ = axy2+ const Substituting this expression into (12.3.2.18) and taking into account relation (12.3.2.12), we find a first integral of equation (12.3.2.17):
yp + axy2= C1, where p = y x .
Setting w = y2, we arrive at the first-order linear equation w x+ 2axw = 2C 1 , which is easy to integrate Thus,
we find the solution of the original equation in the form:
y2= 2C 1 exp –ax2
exp ax2
dx + C2exp –ax2
.
12.3.3 Methods of Regular Series Expansions with Respect to the
Independent Variable
12.3.3-1 Method of expansion in powers of the independent variable
A solution of the Cauchy problem
y
y (x0) = y0, y
can be sought in the form of a Taylor series in powers of the difference (x–x0), specifically:
y (x) = y(x0) + y x (x0)(x – x0) + y xx (x0)
2! (x – x0)
2+ y xxx (x0)
3! (x – x0)
3+· · · (12.3.3.3)
The first two coefficients y(x0) and y x (x0) in solution (12.3.3.3) are defined by the initial
conditions (12.3.3.2) The values of the subsequent derivatives of y at the point x = x0 are determined from equation (12.3.3.1) and its derivative equations (obtained by successive differentiation of the equation) taking into account the initial conditions (12.3.3.2) In
particular, setting x = x0 in (12.3.3.1) and substituting (12.3.3.2), we obtain the value of the second derivative:
y
xx (x0) = f (x0, y0, y1). (12.3.3.4)
... one zero, and hence y ≠ 0forsufficiently large x.
If f (x)≤ 0for all x and f (x)0, then y≡ 0is the only solution bounded for all... k2>0for x → ∞ and f ≥ 0, then the solutions of the equation for large x behave similarly to those of the equation y + k2y=0... more often implicit or parametric forms of the general solution are encountered
12.3.1-2 Cauchy problem The existence and uniqueness theorem
Cauchy problem: Find a solution of