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

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries on Distribution Theory

**Definition**

**1.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Definition**

**2.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

**Corollary**

**3.**

**Lemma**

**5.**

**Proof.**

#### 2.1. Fractional Derivative and Distributions in the Space ${\mathcal{D}}_{R}^{\prime}$

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Lemma**

**6.**

**Condition**

**1.**

**Definition**

**6.**

**Remark**

**1.**

**Lemma**

**7.**

**Lemma**

**8.**

**Lemma**

**9.**

**Proof.**

**Remark**

**2.**

#### 2.2. Fractional Derivative and Distributions in the Space ${\mathcal{D}}_{W}^{\prime}$

**Remark**

**3.**

**Lemma**

**10.**

**Proof.**

**Lemma**

**11.**

**Proof.**

**Corollary**

**4.**

**Proof.**

**Corollary**

**5.**

**Lemma**

**12.**

**Proof.**

**Remark**

**4.**

**Remark**

**5.**

#### 2.3. Some Primitive Leibniz’s formulas

**Lemma**

**13.**

**Proof.**

## 3. Green’s Function for Inhomogeneous Differential Equations with Polynomial Coefficients

**Condition**

**2.**

- (i)
- $f(t)$ is a locally integrable function on ${\mathbb{R}}_{\ge 0}$,
- (ii)
- $f(t)={}_{0}{D}_{R}^{\beta}{f}_{\beta}(t)$ satisfies Condition 1, where ${f}_{\beta}(t)$ is a locally integrable function on ${\mathbb{R}}_{\ge 0}$,
- (iii)
- $f(t)={g}_{\nu}(t)=\frac{1}{\mathrm{\Gamma}(\nu )}{t}^{\nu -1}$, and $\tilde{f}(t)=f(t)\tilde{H}(t)=\widehat{f}(D)\delta (t)={\widehat{g}}_{\nu}(D)\delta (t)={D}^{-\nu}\delta (t)$, for $\nu \in \mathbb{C}\backslash {\mathbb{Z}}_{<1}$.

**Lemma**

**14.**

**Lemma**

**15.**

**Proof.**

**Definition**

**7.**

**Lemma**

**16.**

**Proof.**

**Lemma**

**17.**

**Proof.**

**Lemma**

**18.**

**Proof.**

#### Green’s Function for Inhomogeneous Differential Equations with Constant Coefficients

**Lemma**

**19.**

**Proof.**

**Lemma**

**20.**

**Lemma**

**21.**

**Proof.**

**Remark**

**6.**

## 4. Particular Solution of Kummer’s Differential Equation

#### 4.1. Green’s Function for Kummer’s Differential Equation in which Condition 2(i) Is Satisfied

**Lemma**

**22.**

**Lemma**

**23.**

**Theorem**

**1.**

**Proof.**

**Remark**

**7.**

#### 4.2. Green’s Function for Kummer’s Differential Equation in which Condition 2(ii) Is Satisfied

**Lemma**

**24.**

**Remark**

**8.**

**Theorem**

**2.**

#### 4.3. Particular Solution of Kummer’s Differential Equation in which Condition 2(iii) Is Satisfied

**Lemma**

**25.**

**Proof.**

**Lemma**

**26.**

**Theorem**

**3.**

#### 4.4. Proofs of Lemmas 22–24 and Theorem 2

**Proof of Lemma**

**22.**

**Lemma**

**27.**

**Proof.**

**Proof**

**of**

**Lemma**

**23.**

**Proof**

**of**

**Lemma**

**24.**

**Proof**

**of**

**Theorem**

**2.**

## 5. Particular Solution of the Hypergeometric Differential Equation

#### 5.1. Green’s Function for the Hypergeometric Differential Equation when Condition 2(i) Is Satisfied

**Lemma**

**28.**

**Lemma**

**29.**

**Theorem**

**4.**

**Proof.**

#### 5.2. Green’s Function for the Hypergeometric Differential Equation in which Condition 2(ii) Is Satisfied

**Lemma**

**30.**

**Remark**

**9.**

**Theorem**

**5.**

#### 5.3. Particular Solution of the Hypergeometric Differential Equation in which Condition 2(iii) Is Satisfied

**Lemma**

**31.**

**Proof.**

**Lemma**

**32.**

**Theorem**

**6.**

#### 5.4. Proofs of Lemmas 28–30 and Theorem 5

**Proof**

**of**

**Lemma**

**28.**

**Lemma**

**33.**

**Proof.**

**Proof**

**of**

**Lemma**

**29.**

**Proof**

**of**

**Lemma**

**30.**

**Proof**

**of**

**Theorem**

**5.**

## 6. Solution of Inhomogeneous Differential Equations with Constant Coefficients

**Lemma**

**34.**

**Proof.**

#### 6.1. Solution of an Inhomogeneous Differential Equation of the First Order

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

#### 6.2. Solution of an Inhomogeneous Differential Equation of the Second Order

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

#### 6.3. Application of the Theorems in Section 6.1

**Lemma**

**35.**

**Lemma**

**36.**

**Proof.**

**Lemma**

**37.**

**Proof.**

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Morita, T.; Sato, K.-i.
Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function. *Mathematics* **2017**, *5*, 62.
https://doi.org/10.3390/math5040062

**AMA Style**

Morita T, Sato K-i.
Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function. *Mathematics*. 2017; 5(4):62.
https://doi.org/10.3390/math5040062

**Chicago/Turabian Style**

Morita, Tohru, and Ken-ichi Sato.
2017. "Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function" *Mathematics* 5, no. 4: 62.
https://doi.org/10.3390/math5040062