Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green ’ s Function

The particular solutions of inhomogeneous differential equations with polynomial coefficients in terms of the Green’s function are obtained in the framework of distribution theory. In particular, discussions are given on Kummer’s and the hypergeometric differential equation. Related discussions are given on the particular solution of differential equations with constant coefficients, by the Laplace transform.


Introduction
In the present paper, we treat the problem of obtaining the particular solutions of a differential equation with polynomial coefficients in terms of the Green's function.
In a preceding paper [1], this problem is studied in the framework of distribution theory, where the method is applied to Kummer's and the hypergeometric differential equation. In another paper [2], this problem is studied in the framework of nonstandard analysis, where a recipe of solution of the present problem is presented, and it is applied to a simple fractional and a first-order ordinary differential equation.
In the present paper, we present a compact recipe based on nonstandard analysis, which is obtained by revising the one given in [2]. As an example, it is applied to Kummer's differential equation.
The presentation in this paper follows those in [1,2], in Introduction and in many descriptions in the following sections.
Here Z, R and C are the sets of all integers, all real numbers and all complex numbers, respectively, and Z >a = {n ∈ Z | n > a}, Z <b = {n ∈ Z | n < b} and Z [a,b] In accordance with Definition 1, when u for ν ∈ C\Z <1 and τ ∈ R. Here R D t is used in place of usually used notation τ D R , in order to show that the variable is t.
In [1,6], discussions are made of an ordinary differential equation, which is expressed by (1) for ρ l = n − l, in terms of distribution theory, and with the aid of the analytic continuation of Laplace transform, respectively. In those papers, solutions are given of differential equations with an inhomogeneous term f (t), which satisfies one of the following three conditions.

Green's Function in Distribution Theory
In a recent paper [5], the solution of Euler's differential equation in distribution theory is compared with the solution in nonstandard analysis. In distribution theory [1,[7][8][9], we use distributionH(t), which corresponds to function H(t), differential operator D and distribution δ(t) = DH(t), which is called Dirac's delta function.
When ν ∈ C + and n ∈ Z >0 ,g ν (t) : is a regular distribution, and D ng ν (t) = D n−ν+1H (t) = D n−ν δ(t) is a distribution but is not a regular one, if ν − n ∈ C\C + .
In [1], the following theorem is given.
Theorem 1. Let f (t) satisfy Condition 1 (i) and G 0 (t, τ) be the one given in Lemma 1. Then u f (t) given by is a particular solution of Equation (1).
Proof. By using Equations and (9), we have By taking the derivative of the first and the last member in this equation with respect to t, we confirm that Equation (1) is satisfied by u(t) = u f (t).

Preliminaries on Nonstandard Analysis
In the present paper, we use nonstandard analysis [10], where infinitesimal numbers are used. We denote the set of all infinitesimal real numbers by R 0 . We also use , which is such that if ∈ R 0 >0 and N ∈ Z >0 , then < 1 N . We use R ns , which has subsets R and R 0 . If x ∈ R ns and x / ∈ R, x is expressed as x 1 + by x 1 ∈ R and ∈ R 0 , where x 1 may be 0 ∈ R. Equation x y for x ∈ R ns and y ∈ R ns , is used, when x − y ∈ R 0 . We denote the set of all infinitesimal complex numbers by C 0 , which is the set of complex numbers z which satisfy |Re z| + |Im z|∈ R 0 . We use C ns , which has subsets C and C 0 . If z ∈ C ns and z / ∈ C, z is expressed as z 1 + by z 1 ∈ C and ∈ C 0 , where z 1 may be 0 ∈ C. Remark 5. In nonstandard analysis [10], in addition to infinitesimal numbers, we use unlimited numbers, which are often called infinite numbers. In the present paper, we do not use them, but if we use them, we have to consider sets R ∞ and C ∞ such that if ω ∈ R ∞ , there exists ∈ R 0 satisfying ω = 1 , and if ω ∈ C ∞ , there exists ∈ C 0 satisfying ω = 1 , and then R ns In place of (4), we now use for all ρ ∈ C and ν ∈ C, where ∈ R 0 >0 .
Then, the index law: always holds.
Remark 6. When ∈ R 0 or ∈ C 0 , we often ignore terms of O( ) compared with a term of O( 0 ). For instance, when ν ∈ R >0 and ν − ρ ∈ R >0 , we adopt 1 in place of (12). In the following, we often use "=" in place of " ".
In the present study in nonstandard analysis, ∈ R 0 >0 is used, and H(t) and δ(t) = DH(t), respectively, are replaced by which tends to H(t) in the limit → 0, and by Lemma 3. In the notation in Remark 1, H (t) = g 1+ (t), δ (t) = g (t), and we have

Summary of the Following Sections
In Section 2, a recipe of solution of Equation (1), in nonstandard analysis, is presented. We there consider the solution of the following equation forũ(t): Here, the inhomogeneous terms f (t) andf (t) are assumed to satisfy one of the following four conditions.
f β,0 (t) is locally integrable on R >0 , and c β, is a constant.
(iv)f (t) and f (t) are expressed as follows:

Remark 7. Lemma 3 shows that when Condition
does not always hold, and when Condition 2 (iii) is satisfied, R D tfβ (t) = 0.
In Sections 3 and 4, full expressions of the Green's functions and the solutions, are derived along the recipe given in Section 2, for Kummer's differential equation: where a, b and c are constants satifying a = 0 and b = 0. Section 5 is for Conclusion. In Section 6, a concluding remark is given.

Recipe of Solution of Differential Equation, in Nonstandard Analysis
In obtaining a particular solution of Equation (1) forf (t) satisfying Condition 2 (i), in place of the Green's function defined in Remark 4, we use it defined in the following definition. Definition 2. Letp n, (t, R D t ) be given by Equation (21). Then for ∈ R 0 >0 and τ ∈ R, the Green's function G (t, τ) for Equation (1) satisfies Lemma 5. Let G (t, τ) be defined as in Definition 2, and G 0 (t, τ) := R D t G (t, τ). Then G 0 (t, τ) is a complementary solution of Equation (1) on R >τ , and R D −1 t p n (t, R D t )G 0 (t, τ) = 1 at any value of t satisfying t > τ.
Proof. These are confirmed by applying R D t and R D −1+ t to Equation (25), by noting Lemma 3. Proof. This is confirmed by replacingũ(t) andf (t) byũ c (t) and 0 in Equation (20), and then applying R D t to the equation.

Theorem 2.
Let Condition 2 (i) be satisfied, G (t, τ) and G 0 (t, τ) be given as in Lemma 5. Theñ u f (t) given byũ is the particular solution of Equation (20) for the termf (t), and u f (t) given by consists of the particular solution for the term f (t) and a complementary solution of Equation (1). (27), (25) and (18), we obtaiñ

Proof. By using Equations
which is a proof forũ f (t).

When Condition 2 (iii) is satisfied, Equation (20) is expressed as
Since Condition 2 (iii) is a special case of Condition 2 (ii) in which f β (t) = 0 and c β, = 1, we obtain the following theorem from Theorem 3.

Solution of Kummer's Differential Equation, I
We construct the transformed differential equation of Equation (24), which corresponds to Equation (20). For this purpose, we use the following lemma.

Lemma 9.
Let λ ∈ C + , m ∈ Z >−1 and ρ = m − λ. Then Proof. When m = 0 and ρ = −λ, this is confirmed with the aid of Formula (2), as follows: We prove (38) by mathematical induction. In fact, when (38) holds for a value n ∈ Z >−1 of m, we confirm it to hold even for m = n + 1, by applying d dt to (38).

Remark 9.
When u(t) = t ν+ Γ(ν+ +1) H(t), by using (12), we confirm (38) as follows: With the aid of Formula (38) for ρ = − , we construct the following transformation of Equation (24) forũ(t) = R D − t u(t), which corresponds to Equation (20): When Condition 2 (i) is satisfied, in accordance with Definition 2, we define the Green's function G K, (t, τ), which satisfies for τ ∈ R. The solutions of Equations (39) and (24) are then given with the aid of Theorem 2 and the following lemma.
In the present paper, these equations are proved in Lemmas 11 and 12 given below. Lemma 11. Let K 1 (t) be given by (41). Then G K, (t, 0) and G K,0 (t, 0), given by are a particular solution of Equation (40) for τ = 0, and a complementary solution of Equation (24), respectively.
A proof of the statement for G K, (t, 0) is given in Section 3.1, and the statement for G K,0 (t, 0) is due to Lemma 5. Lemma 12. Let K 2 (t) be given by (42). Thenũ c (t) and u c (t), given bỹ are complementary solutions of Equations (39) and (24), respectively.
A proof of the statement forũ c (t) is given in Section 3.1, and the statement for u c (t) is due to Lemma 6.
The differential equation satisfied by the Green's function G K, (t, τ) for Equation (24) is given by Equation (40).
In obtaining the last term in Equation (64), we use the following formulas: Theorem 7 shows that iff (t) = R D β t δ (t), the particular solution of Equation (39) is given by Equation (62). As a consequence, we have the following theorem.
Condition c − β / ∈ Z <1 in Lemma 14 requires the condition c − β l / ∈ Z <1 for all l ∈ Z >0 , in the present case.
Lemma 17. Lemma 12, Remark 10 and Lemma 6 show thatw c (t) and w c (t), given bỹ are complementary solutions of Equations (57) and (56), respectively, and then Remark 8 shows thatũ c (t) and u c (t), given byũ c (t) = R D β tw c (t) and u c (t) = R D tũ c (t), respectively, are the complementary solutions of Equations (39) and (24), which are given in Lemma 12.

Conclusions
In [1], the problem of obtaining the particular solution of an inhomogeneous ordinary differential equation with polynomial coefficients is discussed in terms of the Green's function, in the framework of distribution theory. It is applied to Kummer's and the hypergeometric differential equation.
In [2], a compact recipe is presented, which is applicable to the case of an inhomogeneous fractional differential equation, which is expressed by Equation (1). In the recipe, the particular solution is given by Theorems 2, 3 or 4, according as the inhomogeneous part satisfies Condition 2 (i), (ii) or (iii), in the framework of nonstandard analysis. It is applied to a simple fractional and an ordinary differential equation.
In Section 2, in the present paper, a compact revised recipe in nonstandard analysis is presented, which is more closely related with distribution theory. In this case, the particular solution is given by Theorems 2, 3, 4 or 5, according as the inhomogeneous part satisfies Condition 2 (i), (ii), (iii) or (iv). In Sections 3 and 4, it is applied to inhomogeneous Kummer's differential Equation (24). In solving Equation (24) in nonstandard analysis, we construct transformed Equation (39) from it. In Section 3, we obtain the solution of Equation (39) by using the Green's function, and obtain the solution of Equation (24) from it. In Section 4, we construct further transformed Equation (57) from Equation (39), obtain the solutions of Equation (57) by using the Green's function, and then obtain the solutions of Equations (39) and (24) from them. In Corollary 1, a nonstandard solution, which involves infinitestimal terms, is presented.
In [11], an ordinary differential equation is expressed in terms of blocks of classified terms. When the equation is expressed by two blocks of classified terms, the complementary solutions are obtained by using Frobenius' method. In Section 3.1, the Green's function and a complementary solution for Equation (39) are presented by using Frobenius' method.
One of reviewers of this paper asked the author to cite papers [12][13][14], which discuss the solutions of fractional differential equations. When the solutions of the differential equations, which are obtained with the aid of distribution theory, are of interest, the solution by using nonstandard analysis will be useful.

Concluding Remark
In the book of [9], Dirac's delta function δ(t) is introduced as a limit of zero width, of a function which has a single peak at t = 0 and unit area, and is defined as a functional. In the present paper, we study problems in nonstandard analysis, by using a function δ (t) which has an infinitesimal width and unit area.
In a preceding paper [1], the problem of obtaining the particular solution of an inhomogeneous ordinary differential equation, is discussed in terms of distribution theory. In another paper [2], we discussed solution of a fractional and a simple ordinary differential equation, in terms of nonstandard analysis by using two functions δ 1 (t) and δ (t) expressed by two infinitesimal numbers 1 and . In the present paper, we proposed a revised recipe in terms of nonstandard analysis, by using the function δ (t) in place of distribution δ(t) in distribution theory. In the present paper, the recipe is applied only to Kummer's differential equation. The application of the present recipe to other differential equations studied in [1,2], will be given in a separate paper in preparation.
The author desires to have a day when we discuss the merit of using two functions δ 1 (t) and δ (t).
Funding: This research received no external funding.