# Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform

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## Abstract

**:**

## 1. Introduction

**Condition A.**

**Remark 1.**

**Condition B.**

**Condition C.**

**Lemma 1.**

**Remark 2.**

**Condition D.**

**Definition 1.**

**Definition 2.**

#### 1.1. Definition of the AC-Laplace Transform

**Condition E.**

**Definition 3.**

**Remark 3.**

#### 1.2. Remarks on Recent Developments

## 2. AC-Laplace Transform

**Lemma 2.**

**Proof**

**Lemma 3.**

**Lemma 4.**

**Proof**

**Lemma 5.**

**Proof**

**Remark 4.**

**Theorem 1.**

#### 2.1. Recipe of Solving Differential Equation with Polynomial Coefficients

**Theorem 2.**

**Theorem 3.**

**Corollary 4.**

#### 2.2. Term-by-Term Operators for $u(t)$ and $\widehat{u}(s)$

**Definition 4.**

**Proposition 5.**

## 3. Operational in the Spaces ${\mathcal{D}}_{R}^{\prime}$ and ${\mathcal{D}}_{r,R}^{\prime}$

#### 3.1. Operational Calculus in the Spaces ${\mathcal{D}}_{R}^{\prime}$

**Definition 5.**

- (i)
- If $\mathrm{Re}\phantom{\rule{0.277778em}{0ex}}\nu >0$ and $u(t)H(t)\in {\mathcal{L}}_{loc}(\mathbb{R})$, then ${D}^{-\nu}\tilde{u}(t)\phantom{\rule{4pt}{0ex}}\u2022\phantom{\rule{-3.69885pt}{0ex}}-\phantom{\rule{-3.69885pt}{0ex}}\circ \phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}[{}_{0}{D}_{R}^{-\nu}u(t)]H(t)$.
- (ii)
- If $h(t)\in {\mathcal{D}}_{R}^{\prime}$, then ${D}^{\lambda}h(t)\in {\mathcal{D}}_{R}^{\prime}$.
- (iii)
- The index law stated in the following lemma is valid.

**Lemma 6.**

**Definition 6.**

**Remark 5.**

**Lemma 7.**

**Lemma 8.**

**Definition 7.**

**Definition 8.**

**Lemma 9.**

**Proof**

#### 3.1.1. Recipe of Solving Linear Differential Equation with Polynomial Coefficients

**Lemma 10.**

**Theorem 6.**

**Proposition 7.**

**Corollary 8.**

#### 3.2. Operational Calculus in the Space ${\mathcal{D}}_{r,R}^{\prime}$

#### 3.3. Distributions in the Space ${\mathcal{D}}_{R}^{\prime}$

**Definition 9.**

**Lemma 11.**

**Definition 10.**

**Lemma 12.**

#### 3.4. Distributions in the Space ${\mathcal{D}}_{r,R}^{\prime}$

**Definition 11.**

**Lemma 13.**

## 4. Solution of Modified Kummer’s DE

#### 4.1. Solution Satisfying $(1-c){u}_{0}=0$

**Lemma 14.**

#### 4.2. Solution Satisfying ${u}_{0}=1$

**Lemma 15.**

#### 4.3. Solution by the Basic Method

#### 4.4. Solution by the Modified Nishimoto’s Method

**Remark 6.**

## 5. Solution of the Hypergeometric DE

**Lemma 16.**

**Proof**

#### 5.1. Solution Satisfying $(1-c){u}_{0}=0$

**Lemma 17.**

**Proof**

**Remark 7.**

**Remark 8.**

#### 5.2. Solution Satisfying ${u}_{0}=1$

**Lemma 18.**

#### 5.3. Solution by the Basic Method

#### 5.4. Solution by the Modified Nishimoto’s Method

## 6. Solution of Bessel’s DE

## 7. Solution of Hermite’s DE

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Morita, T.; Sato, K.-i. Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform. *Mathematics* **2016**, *4*, 19.
https://doi.org/10.3390/math4010019

**AMA Style**

Morita T, Sato K-i. Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform. *Mathematics*. 2016; 4(1):19.
https://doi.org/10.3390/math4010019

**Chicago/Turabian Style**

Morita, Tohru, and Ken-ichi Sato. 2016. "Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform" *Mathematics* 4, no. 1: 19.
https://doi.org/10.3390/math4010019