Nth Order Linear Differential Equations Research Articles

On soluble nth order linear differential equations

The expansion of a product of n linear factors in the operator D is obtained and this leads to a simple solution of the corresponding nth order linear differential equations with variable coefficients.

Solution Spaces of Differential Equations

The solution space of a linear homogeneous differential equation is a vector space. Conversely, given a finite dimensional vector space of analytic functions, it is easy to construct (see below) a differential equation with the given vector space as its solution space. The theorem proved below gives two properties of a finite dimensional vector space of analytic functions, each of which is a necessary and sufficient condition for the corresponding differential equation to be a constant coefficient differential equation. The solution space of the nth order linear homogeneous differential equation with analytic coefficients, d_n dy an(z) dz +*......al(z) + ao(z)y = O, with an(z) O,

Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity

We find analogues of known results on nth order linear differential equations for nth order linear difference equations. These include the concept of disconjugacy, Pólya’s criterion for disconjugacy, Frobenius factorizations, generalized Sturm theorems, existence and properties of principal solutions, signs of Green’s functions, and completely monotone families of solutions of equations depending on a parameter.

Limit circle classification and boundedness of solutions

SynopsisUsing an algebraic approach to the nth order linear differential equations (see [3] forn= 2 and [13and14] forn≧ 2) it is shown the essence of the relation between the limit circle classification and boundedness of solutions ofy″ =q(t)y. On this example it is demonstrated that if a problem can be formulated in the rank of the approach, then it is only a technical matter to answer it, e.g. the relationship studied here is based on a standard fact from the theory of functions.

Nonoscillatory solutions of a class of nth-order linear differential equations

Nonoscillatory solutions of a class of nth-order linear differential equations

Nonoscillation and eventual disconjugacy

If every solution of an nth order linear differential equation has only a finite number of zeros in [ 0 , ∞ ) [0,\infty ) , it is not generally true that for sufficiently large c , c > 0 c,c > 0 , every solution has at most n − 1 n - 1 zeros in [ c , ∞ ) [c,\infty ) . Settling a known conjecture, we show that for any n, the above implication does hold for a special type of equation, L n y + p ( x ) y = 0 {L_n}y + p(x)y = 0 , where L n {L_n} is an nth order disconjugate differential operator and p ( x ) p(x) is a continuous function of a fixed sign.

Comparison of Eigenvalues Associated With Linear Differential Equations of Arbitrary Order

Existence and comparison theorems for eigenvalues of $(k,n - k)$-focal point and $(k,n - k)$-conjugate point problems are proved for a class of nth order linear differential equations for arbitrary n.

Asymptotic Integration of Linear Differential Equations Subject to Integral Smallness Conditions Involving Ordinary Convergence

The problem of asymptotic behavior of solutions of an nth order linear differential equation is reconsidered, and a result obtained by Hartman under integral smallness conditions requiring absolute integrability is shown to hold with most of the conditions stated in terms of ordinary integrability. Results of Fubini and Halanay for linear perturbations of nonoscillatory second order equations are similarly extended.

Approximation with sums of exponentials in Lp[0, ∞)

We consider the problem of approximating a given f from L p [0, ∞) by means of the family V n ( S) of exponential sums; V n ( S) denotes the set of all possible solutions of all possible nth order linear homogeneous differential equations with constant coefficients for which the roots of the corresponding characteristic polynomials all lie in the set S. We establish the existence of best approximations, show that the distance from a given f to V n ( S) decreases to zero as n becomes infinite, and characterize such best approximations with a first-order necessary condition. In so doing we extend previously known results that apply in L p [0, 1].

Existence and comparison of eigenvalues of 𝑛th order linear differential equations

Existence and comparison of eigenvalues of 𝑛th order linear differential equations

Comparison of eigenvalues associated with linear differential equations of arbitrary order

Existence and comparison theorems for eigenvalues of ( k , n − k ) (k,n - k) -focal point and ( k , n − k ) (k,n - k) -conjugate point problems are proved for a class of nth order linear differential equations for arbitrary n.

On the Continuity of Conjugate Point Functions

With the kth conjugate point of t$\eta _k (t)$, defined for a given nth order linear differential equation as the minimum number r greater than t such that there exists a nontrivial solution of the differential equation having zeros at t and r and at least $n + k - 1$ zeros on the interval $[ t,r ]$, it is well known that $\eta _k (t)$ is a strictly increasing continuous function of t when $k = 1$; however, this question remains open when $k > 1$. For $k\geqq 1$, the main theorems of this paper give conditions which guarantee that $\eta _k (t)$ is strictly increasing and continuous and that $\eta _k (t)$ depends continuously on the coefficients in the given differential equation. The relationship between these results and previous results for fourth order differential equations is indicated.

On the zeros of solutions of Nth order linear differential equations

where -a 0, U(X) is said to have a zero at x = t. To say that y(x) has a simple zero, a double zero, or a triple zero at x = t means that [t; y] = 1, [t; y] = 2, or [t; y] = 3, respectively. A reference to the number of zeros

Interpolation Between Consecutive Conjugate Points of an nth Order Linear Differential Equation

Interpolation Between Consecutive Conjugate Points of an nth Order Linear Differential Equation

Interpolation between consecutive conjugate points of an 𝑛th order linear differential equation

The interpolation problem x ( n ) + P n − 1 x ( n − 1 ) + ⋯ + P 0 x = 0 {x^{(n)}} + {P_{n - 1}}{x^{(n - 1)}} + \cdots + {P_0}x = 0 , x ( i ) ( t j ) = 0 , i = 0 , ⋯ , k j − 1 , j = 0 , ⋯ , m {x^{(i)}}({t_j}) = 0,i = 0, \cdots ,{k_j} - 1,j = 0, \cdots ,m , is studied on the conjugate interval [ a , η 1 ( a ) ] [a,{\eta _1}(a)] . The main result is that there exists an essentially unique nontrivial solution of the problem almost everywhere, provided k 1 + ⋯ + k m ≥ n {k_1} + \cdots + {k_m} \geq n , and cer tain other inequalities are satisfied, with a = t 0 > t 1 > ⋯ > t m = η 1 ( a ) a = {t_0} > {t_1} > \cdots > {t_m} = {\eta _1}(a) . In particular, this paper corrects the results of Azbelev and Caljuk (Mat. Sb. 51 (93) (1960), 475-486; English transl., Amer. Math. Soc. Transl. (2) 42 (1964), 233-245) on third order equations, and shows that their results are correct almost everywhere.

On the Extremal Solutions of nth-Order Linear Differential Equations

On the Extremal Solutions of nth-Order Linear Differential Equations

On the extremal solutions of 𝑛th-order linear differential equations

Distribution of zeros of extremal solutions of linear nth-order differential equations is discussed. Existence and nonexistence of extremal solutions with certain zero distributions are established. For instance, it is proved that every extremal solution for [ α , η 1 ( α ) ] [\alpha , {\eta _1}(\alpha )] of the equation y ( n ) + p n − 1 y ( n − 1 ) + ⋯ + p 0 y = 0 {y^{(n)}} + {p_{n - 1}}{y^{(n - 1)}} + \cdots + {p_0}y = 0 has a zero of order 2 at η 1 ( α ) {\eta _1}(\alpha ) and has no more than n − 2 n - 2 zeros on [ α , η 1 ( α ) ) if p i ≦ 0 , i = 0 , 1 , ⋯ , n − 2 [\alpha , {\eta _1}(\alpha ))\;{\text {if}}\;{p_i} \leqq 0,i = 0,1, \cdots , n - 2 .

Pointwise Bounds on Derivatives of Solutions to Ordinary Differential Equations

Various pointwise estimates on the derivatives of solutions to nth order linear differential equations in terms of an associated Taylor polynomial are derived. These estimates are used to obtain a necessary and sufficient condition for a function which satisfies a sequence of linear differential equations on an interval to be regular on that inverval.

Simple Zeros of Solutions of nth-Order Linear Differential Equations

Simple Zeros of Solutions of nth-Order Linear Differential Equations

Simple zeros of solutions of 𝑛𝑡ℎ-order linear differential equations

Let the n n th-order linear differential equation L y = 0 Ly = 0 have a nontrivial solution with n n zeros (counting multiplicities) on an interval [ α , β ] [\alpha ,\beta ] . A condition under which L y = 0 Ly = 0 has a solution with n n simple zeros on [ α , β ] [\alpha ,\beta ] is established. Also, a new proof is given for a known result concerning an interval of the type [ α , β ) [\alpha ,\beta ) .