# On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach

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## Abstract

**:**

## 1. Introduction

## 2. The Advanced Equation: $\mathbf{0}\le \mathit{t}\le \mathit{\tau}/\mathbf{2}$

#### 2.1. Reduced Model: $\alpha =0$

#### 2.2. Full Model: $\alpha \ne 0$

## 3. Existence and Uniqueness

## 4. Compact Form for the Periodic Solution of the Advanced Equation

**Theorem**

**1.**

**Proof.**

## 5. Special Cases of the Advanced Equation

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Lemma**

**6.**

**Proof.**

## 6. Characteristics of the Solution of the Advanced Equation

#### 6.1. Behavior of the Periodic Solution

#### 6.2. The Amplitude, Phase, and Critical Values of the Advance Parameter $\tau $: ${\tau}_{c}$

#### 6.3. Behavior of the Polynomial Solution

## 7. The Delay Equation: $\mathit{t}\ge \mathit{\tau}/\mathbf{2}$

#### 7.1. The Solution in the Interval: $\tau /2\le t\le \tau $

#### 7.2. The Solution in the Interval of $t>\tau $

## 8. Properties and Behavior of the Solution in the Full Domain

## 9. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Behavior of the solution at different values of $\alpha $ and $\beta $ when $\lambda =1$ and $\tau =4$.

**Figure 6.**The periodic solution (47) for the scalar model (1) at $\lambda =1$ and some selected values of $\alpha $ and $\beta $ ($(\alpha ,\beta )=(\pm 1,\pm 2)$) in the domain of one period ($0\le t\le 2\pi /\sqrt{3}$).

**Figure 7.**The periodic solution (47) for the scalar model (1) at $\lambda =1$ and some selected values of $\alpha $ and $\beta $ ($(\alpha ,\beta )=(\pm \sqrt{2},\pm \sqrt{3})$) in the domain of one period ($0\le t\le 2\pi $).

**Figure 8.**The periodic solution (47) for the scalar model (1) at $\lambda =1$ and some selected values of $\alpha $ and $\beta $ ($(\alpha ,\beta )=(\pm 2,\pm 5)$) in the domain of two periods ($0\le t\le 4\pi /\sqrt{5}$).

**Figure 9.**Behavior of the polynomial solution at different values of $\alpha >0$ when $\tau =4$ and $\lambda =1$.

**Figure 10.**Behavior of the polynomial solution at different values of $\alpha <0$ when $\tau =4$ and $\lambda =1$.

**Figure 11.**Plot of the solution in the interval of $[0,\tau /2]$ for the advanced equation (blue curve) and in the intervals of $[\tau /2,\tau ]$ (red curve) and $(\tau ,\infty )$ (green curve) for the delay equation. The black dots represent the connection points between the solutions at $\lambda =1$, $\alpha =1$, and $\beta =2$ when $\tau =6$.

**Figure 12.**Plot of the solution in the interval of $[0,\tau /2]$ for the advanced equation (blue curve) and in the intervals of $[\tau /2,\tau ]$ (red curve) and $(\tau ,\infty )$ (green curve) for the delay equation. The black dots represent the connection points between the solutions at $\lambda =1$, $\alpha =1$, and $\beta =2$ when $\tau =12$.

**Figure 13.**Plot of the solution in the interval of $[0,\tau /2]$ for the advanced equation (blue curve) and in the intervals of $[\tau /2,\tau ]$ (red curve) and $(\tau ,\infty )$ (green curve) for the delay equation. The black dots represent the connection points between the solutions at $\lambda =1$, $\alpha =-2$, and $\beta =3$ when $\tau =6$.

**Figure 14.**Plot of the solution in the interval of $[0,\tau /2]$ for the advanced equation (blue curve) and in the intervals of $[\tau /2,\tau ]$ (red curve) and $(\tau ,\infty )$ (green curve) for the delay equation. The black dots represent the connection points between the solutions at $\lambda =1$, $\alpha =-2$, and $\beta =3$ when $\tau =12$.

**Figure 15.**Plot of ${\varphi}^{\prime}\left(t\right)$ in the intervals of $[0,\tau /2]$ (blue curve), $[\tau /2,\tau ]$ (red curve), and $(\tau ,\infty )$ (green curve) at $\lambda =1$, $\alpha =1$, and $\beta =2$ when $\tau =6$.

**Figure 16.**Plot of ${\varphi}^{\prime}\left(t\right)$ in the intervals of $[0,\tau /2]$ (blue curve), $[\tau /2,\tau ]$ (red curve), and $(\tau ,\infty )$ (green curve) at $\lambda =1$, $\alpha =1$, and $\beta =2$ when $\tau =12$.

**Figure 17.**Plot of ${\varphi}^{\prime}\left(t\right)$ in the intervals of $[0,\tau /2]$ (blue curve), $[\tau /2,\tau ]$ (red curve), and $(\tau ,\infty )$ (green curve) at $\lambda =1$, $\alpha =-2$, and $\beta =3$ when $\tau =6$.

**Figure 18.**Plot of ${\varphi}^{\prime}\left(t\right)$ in the intervals of $[0,\tau /2]$ (blue curve), $[\tau /2,\tau ]$ (red curve), and $(\tau ,\infty )$ (green curve) at $\lambda =1$, $\alpha =-2$, and $\beta =3$ when $\tau =12$.

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**MDPI and ACS Style**

Alshomrani, N.A.M.; Ebaid, A.; Aldosari, F.; Aljoufi, M.D.
On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach. *Axioms* **2024**, *13*, 129.
https://doi.org/10.3390/axioms13020129

**AMA Style**

Alshomrani NAM, Ebaid A, Aldosari F, Aljoufi MD.
On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach. *Axioms*. 2024; 13(2):129.
https://doi.org/10.3390/axioms13020129

**Chicago/Turabian Style**

Alshomrani, Nada A. M., Abdelhalim Ebaid, Faten Aldosari, and Mona D. Aljoufi.
2024. "On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach" *Axioms* 13, no. 2: 129.
https://doi.org/10.3390/axioms13020129