Integral Transforms and the Hyers–Ulam Stability of Linear Differential Equations with Constant Coefficients

Abstract

Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations, including Laplace, Kamal, Tarig, Aboodh, Mahgoub, Sawi, Fourier, Shehu, and Elzaki integral transforms. This work provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam stability of differential equations with constant coefficients, utilizing the Kamal transform, where we focus on first- and second-order linear equations. In particular, in this work, we employ the Kamal transform to determine the Hyers–Ulam stability and Hyers–Ulam stability constants for first-order complex constant coefficient differential equations and, for second-order real constant coefficient differential equations, improving previous results obtained by using the Kamal transform. In a section of examples, we compare and contrast our results favorably with those established in the literature using means other than the Kamal transform.

1. Introduction

In the case that a differential equation has an approximate solution, say one obtained through a given numerical scheme, one would want to know whether an actual solution to the differential equation exists that the approximate solution remains close to in some sense. This type of concern is often referred to as the Hyers–Ulam stability question for that differential equation. Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations. The first paper of this type is by Rezaei, Jung, and Rassias [1] using the Laplace transform; other papers using the Laplace transform to investigate Hyers–Ulam stability include [2,3]. Since then, scholars have used a variety of integral transform definitions to explore Hyers–Ulam stability, including the Kamal [4], Mahgoub [5,6,7,8], Tarig [9], Shehu [10], Sawi [11], Aboodh [12,13,14], Fourier [15,16,17,18], general [19], and Elzaki [20] integral transforms. Fourier transforms require functions and equations to be defined on the whole real line $(-\infty ,\infty )$, while the other transforms listed above require functions and equations to be defined on the half-line $[0,\infty )$. In this work, we focus on the half-line equations and methods.

A key motivation of this work is to compare the results on Hyers–Ulam stability (HUS) derived from integral transform methods with those derived using other techniques, such as in [21,22], often through carefully chosen examples. Another motivation is to bridge some of the gaps that may exist in the results between using one method versus another. A third motivation is to discuss the best (minimal) Hyers–Ulam stability constant where appropriate, and another motivation is to explicitly point out instances where a differential equation is not Hyers–Ulam-stable, known as instability results. To this end, Section 2 provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam stability of differential equations with constant coefficients, in particular using the Kamal transform on differential equations with complex constant coefficients of order one with the best HUS constant and real constant coefficients of order two. Section 3 provides several key examples of second-order equations with constant coefficients, and in particular compares and contrasts the results derived here with those found in the available literature, to bridge gaps where possible. In particular, the best Hyers–Ulam stability constant is often discerned for these equations. We utilize the Kamal transform to obtain new and improved results for that particular approach, and as a representative vehicle for this discussion.

2. Main Results

In this section, we consider the first-order linear constant-coefficient homogeneous differential equation

$$p^{\prime }\left(t\right)+\mu p\left(t\right)=0,\mu \in \mathbb{C},t\in (0,\infty ),$$

(1)

and the second-order linear constant-coefficient homogeneous differential equation

$$a{p}^{″}\left(t\right)+b{p}^{\prime }\left(t\right)+cp\left(t\right)=0,a,b,c\in \mathbb{R},t\in (0,\infty ),$$

(2)

with $a\ne 0$. Our main tool to determine the Hyers–Ulam stability of (1), and (2) is the Kamal transform, where we define the Kamal transform and Hyers–Ulam stability forthwith.

Definition 1. 

For a function $z:(0,\infty )\to \mathbb{C}$ such that $\left|z\left(t\right)\right|\le A{e}^{tB}$ for all $t>T$ for some $A,B,T\in (0,\infty )$, define the Kamal transform as

$$\mathcal{K}\left\{z\left(t\right)\right\}={\int }_0^{\infty }z\left(t\right)e^{-\frac{t}{s}}dt=Z\left(s\right),s>0.$$

(3)

If $Z\left(s\right)=\mathcal{K}\left\{z\right(t\left)\right\}$, then the inverse Kamal transform is denoted by

$${\mathcal{K}}^{-1}\left\{Z\left(s\right)\right\}=z\left(t\right).$$

Definition 2. 

Equation (1) is Hyers–Ulam-stable on $I=(0,\infty )$ if there exists a constant $K>0$ with the following property:

For any $\epsilon >0$, and for any continuously differentiable function $\varphi :I\to \mathbb{C}$ satisfying

$$\underset{t\in I}{sup}|{\varphi }^{\prime }\left(t\right)+\mu \varphi \left(t\right)|\le \epsilon ,\mu \in \mathbb{C}$$

there exists a solution $p:I\to \mathbb{C}$ of (1) such that

$$\underset{t\in I}{sup}|\varphi \left(t\right)-p\left(t\right)|\le K\epsilon .$$

We call such K a Hyers–Ulam constant for (1) on I. Likewise, (2) is Hyers–Ulam-stable on $I=(0,\infty )$ if there exists a constant $K>0$ with the following property:

For any $\epsilon >0$, and for any continuously differentiable function $\varphi :I\to \mathbb{C}$ satisfying

$$\underset{t\in I}{sup}|a{\varphi }^{″}\left(t\right)+b{\varphi }^{\prime }\left(t\right)+c\varphi \left(t\right)|\le \epsilon ,$$

there exists a solution $p:I\to \mathbb{C}$ of (2) such that

$$\underset{t\in I}{sup}|\varphi \left(t\right)-p\left(t\right)|\le K\epsilon .$$

According to [4] (Theorem 2.1), Equation (1) has Hyers–Ulam stability for any constant $\mu \in \mathbb{C}$. We prove in the next lemma, using the same Kamal transform method (3) as in [4], that this is incorrect if $\mu =i\beta$, that is, if the constant $\mu \in \mathbb{C}$ in (1) is purely imaginary.

Lemma 1. 

Let $\mu \in \mathbb{C}$. If $\mu =i\beta$ for $\beta \in \mathbb{R}$, then (1) is not Hyers–Ulam-stable.

Proof. 

Assume $p:[0,\infty )\to \mathbb{C}$ such that $\left|p\left(t\right)\right|\le A{e}^{tB}$ for all $t>0$ for some $A,B\in \mathbb{R}$, and let $P\left(s\right)=\mathcal{K}\left\{p\right(t\left)\right\}$, the Kamal transform of p. Taking the Kamal transform of (1) via (3), we obtain

$$\frac{1}{s}P\left(s\right)-p\left(0\right)+i\beta P\left(s\right)=0.$$

Solving this for $P\left(s\right)$ yields $P\left(s\right)=\frac{sp\left(0\right)}{1+i\beta s}$. Taking the inverse Kamal transform of both sides yields

$${\mathcal{K}}^{-1}\left\{P\left(s\right)\right\}=p\left(t\right)=p\left(0\right)e^{-i\beta t},$$

which is the general solution of (1) in this case with $\mu \in \mathbb{C}$ with $\mu =i\beta$. Given an arbitrary $\epsilon >0$, consider the related equation

$${\varphi }^{\prime }\left(t\right)+i\beta \varphi \left(t\right)=\epsilon e^{-i\beta t},t>0.$$

(4)

Note that a solution $\varphi$ satisfies ${sup}_{t\in I}|{\varphi }^{\prime }\left(t\right)+i\beta \varphi \left(t\right)|=\epsilon$, as required in the definition of Hyers–Ulam stability. Assume $\Phi \left(s\right)=\mathcal{K}\left\{\varphi \right(t\left)\right\}$ exists, and take the Kamal transform of both sides of (4) via (3) to obtain

$$\frac{1}{s}\Phi \left(s\right)-\varphi \left(0\right)+i\beta \Phi \left(s\right)=\epsilon \frac{s}{1+i\beta s}.$$

Solving this for $\Phi \left(s\right)$ yields $\Phi \left(s\right)=\frac{s\varphi \left(0\right)}{1+i\beta s}+\frac{s^2\epsilon }{{(1+i\beta s)}^2}$. Taking the inverse Kamal transform of both sides yields

$${\mathcal{K}}^{-1}\{\Phi \left(s\right)\}=\varphi \left(t\right)=\varphi \left(0\right)e^{-i\beta t}+\epsilon t{e}^{-i\beta t},$$

which is the general solution of (4). Then, we have

$$\underset{t\in I}{sup}|\varphi \left(t\right)-p\left(t\right)|=\underset{t\in I}{sup}\left|(\varphi \left(0\right)-p\left(0\right))e^{-i\beta t}+\epsilon t{e}^{-i\beta t}\right|=\underset{t\in I}{sup}\left|\left(\varphi \right(0)-p(0\left)\right)+\epsilon t\right|=\infty$$

for $I=(0,\infty )$. Therefore, we see that (1) is not Hyers–Ulam-stable if $\mu \in \mathbb{C}$ with $\mu =i\beta$. □

Lemma 2. 

Let $\mu \in \mathbb{C}$. If $Re\left(\mu \right)\ne 0$, then the minimal Hyers–Ulam stability constant for (1) is at least $K=\frac{1}{|Re(\mu \left)\right|}$.

Proof. 

Given an arbitrary $\epsilon >0$ and $\mu \in \mathbb{C}$ with $Re\left(\mu \right)\ne 0$, set $\varphi \left(t\right)=\frac{\epsilon }{Re\left(\mu \right)}{e}^{-itIm\left(\mu \right)}$, where $Im\left(\mu \right)$ is the imaginary part of $\mu \in \mathbb{C}$. It is easy to verify that this $\varphi$ satisfies ${\varphi }^{\prime }\left(t\right)+\mu \varphi \left(t\right)=\epsilon e^{-itIm\left(\mu \right)}$, with $|\epsilon e^{-itIm\left(\mu \right)}|\equiv \epsilon$ for all $t\in I$, so that $\varphi$ satisfies (5). Since $p\left(t\right)\equiv 0$ is a solution of (1), we have

$$\underset{t\in I}{sup}|\varphi \left(t\right)-p\left(t\right)|=\underset{t\in I}{sup}\left|\frac{\epsilon }{Re\left(\mu \right)}{e}^{-itIm\left(\mu \right)}\right|=\frac{\epsilon }{|Re(\mu \left)\right|}.$$

Consequently, the best (minimal) possible Hyers–Ulam stability constant for (1) is at least $K=\frac{1}{|Re(\mu \left)\right|}$. □

In [4] (Theorem 2.1), in the case that Equation (1) has Hyers–Ulam stability, the Hyers–Ulam constant K is also incorrect. The following lemma gives the correct Hyers–Ulam stability condition and Hyers–Ulam stability constant. The results also improve those given in [23] for the case of constant $\mu \in \mathbb{C}$.

Lemma 3. 

Let $Re\left(\mu \right)$ be the real part of $\mu \in \mathbb{C}$. If $Re\left(\mu \right)\ne 0$ for $\mu \in \mathbb{C}$, then (1) is Hyers–Ulam-stable with the Hyers–Ulam constant $K=\frac{1}{|Re(\mu \left)\right|}$.

Proof. 

According to the previous lemma, Equation (1) is not Hyers–Ulam-stable if $Re\left(\mu \right)=0$, so throughout this proof, assume $Re\left(\mu \right)\ne 0$. Taking the Kamal transform of (1) via (3), we obtain

$$\frac{1}{s}P\left(s\right)-p\left(0\right)+\mu P\left(s\right)=0,s>0.$$

Solving this for $P\left(s\right)$ yields $P\left(s\right)=\frac{sp\left(0\right)}{1+\mu s}$ for $s\ne -1/\mu$. Taking the inverse Kamal transform of both sides yields

$${\mathcal{K}}^{-1}\left\{P\left(s\right)\right\}=p\left(t\right)=p\left(0\right)e^{-\mu t},$$

which is the general solution of (1). Given an arbitrary $\epsilon >0$, consider the related equation

$${\varphi }^{\prime }\left(t\right)+\mu \varphi \left(t\right)=q\left(t\right),\underset{t\in I}{sup}\left|q\left(t\right)\right|\le \epsilon .$$

(5)

Note that a solution $\varphi$ satisfies ${sup}_{t\in I}|{\varphi }^{\prime }\left(t\right)+\mu \varphi \left(t\right)|\le \epsilon$, as required in the definition of Hyers–Ulam stability. Assume $\Phi \left(s\right)=\mathcal{K}\left\{\varphi \right(t\left)\right\}$ exists, and take the Kamal transform of both sides of (5) via (3) to obtain

$$\frac{1}{s}\Phi \left(s\right)-\varphi \left(0\right)+\mu \Phi \left(s\right)=Q\left(s\right),Q\left(s\right):=\mathcal{K}\left\{q\left(t\right)\right\},s>0.$$

Solving this for $\Phi \left(s\right)$ yields $\Phi \left(s\right)=\frac{s\varphi \left(0\right)}{1+\mu s}+\left(\frac{s}{1+\mu s}\right)Q\left(s\right)$ for $s\ne -1/\mu$. Taking the inverse Kamal transform of both sides yields

$${\mathcal{K}}^{-1}\{\Phi \left(s\right)\}=\varphi \left(t\right)=\varphi \left(0\right)e^{-\mu t}+{\int }_0^t{e}^{-\mu (t-s)}q\left(s\right)ds,$$

which is the general solution of (5). If $Re\left(\mu \right)>0$, let p be the solution of (1) with $p\left(0\right)=\varphi \left(0\right)$. Then we have

$$\begin{array}{cc}\hfill \underset{t\in I}{sup}|\varphi \left(t\right)-p\left(t\right)|& =\underset{t\in I}{sup}\left|{\int }_0^t{e}^{-\mu (t-s)}q\left(s\right)ds\right|\le \epsilon \underset{t\in I}{sup}{e}^{-tRe\left(\mu \right)}{\int }_0^t{e}^{sRe\left(\mu \right)}ds\hfill \\ \hfill & =\epsilon \underset{t\in I}{sup}{e}^{-tRe\left(\mu \right)}\left(\frac{e^{tRe\left(\mu \right)}-1}{Re\left(\mu \right)}\right)=\left(\frac{1}{Re\left(\mu \right)}\right)\epsilon .\hfill \end{array}$$

Therefore, we see that (1) is Hyers–Ulam-stable if $Re\left(\mu \right)>0$, with the Hyers–Ulam constant $K=\frac{1}{Re\left(\mu \right)}$.

Now, assume $Re\left(\mu \right)<0$. Note that, if $\varphi$ is a solution of (5), then

$$\underset{t\to \infty }{lim}\varphi \left(t\right)e^{\mu t}=\varphi \left(0\right)+{\int }_0^{\infty }{e}^{\mu s}q\left(s\right)ds=:p\left(0\right)$$

exists as a finite number. Let p be the solution of (1) with $p\left(0\right)=\varphi \left(0\right)+{\int }_0^{\infty }{e}^{\mu s}q\left(s\right)ds$. Then, we have

$$\begin{array}{cc}\hfill \underset{t\in I}{sup}|\varphi \left(t\right)-p\left(t\right)|& =\underset{t\in I}{sup}\left|-{\int }_t^{\infty }{e}^{-\mu (t-s)}q\left(s\right)ds\right|\le \epsilon \underset{t\in I}{sup}{e}^{-tRe\left(\mu \right)}{\int }_t^{\infty }{e}^{sRe\left(\mu \right)}ds\hfill \\ \hfill & =\epsilon \underset{t\in I}{sup}{e}^{-tRe\left(\mu \right)}\left(\frac{0-1}{Re\left(\mu \right)}\right)=\left(\frac{-1}{Re\left(\mu \right)}\right)\epsilon \hfill \end{array}$$

for $Re\left(\mu \right)<0$. Therefore, we see that (1) is Hyers–Ulam-stable if $Re\left(\mu \right)<0$, with the Hyers–Ulam constant $K=\frac{1}{\left|Re\left(\mu \right)\right|}$. Thus, if $Re\left(\mu \right)\ne 0$, then (1) is Hyers–Ulam-stable with the Hyers–Ulam constant $K=\frac{1}{\left|Re\left(\mu \right)\right|}$. □

Theorem 1. 

Equation (1) is Hyers–Ulam-stable if and only if $Re\left(\mu \right)\ne 0$, with best Hyers–Ulam constant $K=\frac{1}{|Re(\mu \left)\right|}$.

Proof. 

The results follow immediately from Lemmas 1–3. □

We now consider the second-order Equation (2) and determine its Hyers–Ulam stability. In the case of stability, we also find a Hyers–Ulam stability constant.

Theorem 2. 

If ${\lambda }_1,{\lambda }_2\in \mathbb{C}$ are the roots of the characteristic equation

$$a{\lambda }^2+b\lambda +c=0,a\in \mathbb{R}\setminus \left\{0\right\},b,c\in \mathbb{R},$$

(6)

then Equation (2) is Hyers–Ulam-stable if and only if $Re\left({\lambda }_1\right)Re\left({\lambda }_2\right)\ne 0$, where $Re\left(\lambda \right)$ is the real part of λ. In particular, if ${\lambda }_1,{\lambda }_2\in \mathbb{R}$ satisfy ${\lambda }_1{\lambda }_2\ne 0$, then Equation (2) is Hyers–Ulam-stable with the Hyers–Ulam constant $K=\frac{1}{|a{\lambda }_1{\lambda }_2|}$. If ${\lambda }_1,{\lambda }_2\in \mathbb{C}$ satisfy ${\lambda }_1,{\lambda }_2=\alpha \mp \beta i$ with $\alpha \ne 0$ and $\beta >0$, then Equation (2) is Hyers–Ulam-stable with the Hyers–Ulam constant $K=\frac{1}{c}coth\left(\frac{\left|b\right|\pi }{2\sqrt{4ac-b^2}}\right)$.

Proof. 

We divide the proof into Case 1: ${\lambda }_1\ne {\lambda }_2$, and Case 2: ${\lambda }_1={\lambda }_2$.

Case 1: Let ${\lambda }_1,{\lambda }_2$ satisfy (6) with ${\lambda }_1\ne {\lambda }_2$. Assume $p\left(t\right)$ has a Kamal transform $\mathcal{K}\left\{p\right(t\left)\right\}=P\left(s\right)$. Then,

$$\mathcal{K}\left\{p^{″}\left(t\right)\right\}=\mathcal{K}\left\{{\left(p^{\prime }\right)}^{\prime }\left(t\right)\right\}=\frac{1}{s^2}P\left(s\right)-\frac{1}{s}p\left(0\right)-p^{\prime }\left(0\right),s>0,$$

so that the Kamal transform of (2) for $s>0$ is

$$\frac{a}{s^2}P\left(s\right)-\frac{a}{s}p\left(0\right)-a{p}^{\prime }\left(0\right)+\frac{b}{s}P\left(s\right)-bp\left(0\right)+cP\left(s\right)=0,s>0.$$

Solving this for $P\left(s\right)$ with $s>0$ and $s\ne 1/{\lambda }_j$ for $j=1,2$ yields

$$P\left(s\right)=\frac{\left(\frac{a}{s}+b\right)p\left(0\right)+a{p}^{\prime }\left(0\right)}{\frac{a}{s^2}+\frac{b}{s}+c}=\frac{\frac{p\left(0\right)}{s}+\frac{bp\left(0\right)}{a}+p^{\prime }\left(0\right)}{\left(\frac{1}{s}-{\lambda }_1\right)\left(\frac{1}{s}-{\lambda }_2\right)}=\frac{A\left(p\right)}{\left(\frac{1}{s}-{\lambda }_1\right)}+\frac{B\left(p\right)}{\left(\frac{1}{s}-{\lambda }_2\right)},$$

where

$$A\left(p\right)=\frac{\left({\lambda }_1+\frac{b}{a}\right)p\left(0\right)+p^{\prime }\left(0\right)}{{\lambda }_1-{\lambda }_2}\mathrm{and}B\left(p\right)=\frac{\left({\lambda }_2+\frac{b}{a}\right)p\left(0\right)+p^{\prime }\left(0\right)}{{\lambda }_2-{\lambda }_1}.$$

Using the inverse Kamal transform on $P\left(s\right)$ gives us the solution

$$p\left(t\right)=A\left(p\right)e^{{\lambda }_{1}t}+B\left(p\right)e^{{\lambda }_{2}t}$$

for (2). Given arbitrary $\epsilon >0$, suppose $\varphi :I\to \mathbb{R}$ satisfies

$$a{\varphi }^{″}\left(t\right)+b{\varphi }^{\prime }\left(t\right)+c\varphi \left(t\right)=q\left(t\right),\mathrm{where}\underset{t\in I}{sup}\left|q\left(t\right)\right|\le \epsilon .$$

Then, the Kamal transform and basic algebra yield

$$\Phi \left(s\right)=\frac{A\left(\varphi \right)}{\left(\frac{1}{s}-{\lambda }_1\right)}+\frac{B\left(\varphi \right)}{\left(\frac{1}{s}-{\lambda }_2\right)}+\frac{Q\left(s\right)}{a\left(\frac{1}{s}-{\lambda }_1\right)\left(\frac{1}{s}-{\lambda }_2\right)},s>0,s\ne \frac{1}{{\lambda }_1},\frac{1}{{\lambda }_2},$$

so that the inverse Kamal transform gives us

$$\varphi \left(t\right)=A\left(\varphi \right)e^{{\lambda }_{1}t}+B\left(\varphi \right)e^{{\lambda }_{2}t}+\frac{1}{a({\lambda }_2-{\lambda }_1)}{\int }_0^{t}q\left(s\right)\left(e^{{\lambda }_2(t-s)}-e^{{\lambda }_1(t-s)}\right)ds.$$

(7)

Case 1 (i): Let ${\lambda }_1,{\lambda }_2$ satisfy (6) with ${\lambda }_2>{\lambda }_1>0$. Let

$$p\left(t\right)=A\left(\varphi \right)e^{{\lambda }_{1}t}+B\left(\varphi \right)e^{{\lambda }_{2}t}+\frac{1}{a({\lambda }_2-{\lambda }_1)}{\int }_0^{\infty }q\left(s\right)\left(e^{{\lambda }_2(t-s)}-e^{{\lambda }_1(t-s)}\right)ds,$$

which exists and is well defined because ${\lambda }_2>{\lambda }_1>0$ and is a solution of (2). Using (7) and recalling that $\left|q\right(t\left)\right|\le \epsilon$, we have

$$\begin{array}{cc}\hfill \underset{t\in I}{sup}|\varphi \left(t\right)-p\left(t\right)|& =\underset{t\in I}{sup}\frac{1}{\left|a\right|({\lambda }_2-{\lambda }_1)}\left|-{\int }_t^{\infty }q\left(s\right)\left(e^{{\lambda }_2(t-s)}-e^{{\lambda }_1(t-s)}\right)ds\right|\hfill \\ \hfill & \le \underset{t\in I}{sup}\frac{\epsilon }{\left|a\right|({\lambda }_2-{\lambda }_1)}{\int }_t^{\infty }\left|e^{{\lambda }_2(t-s)}-e^{{\lambda }_1(t-s)}\right|ds\hfill \\ \hfill & =\frac{\epsilon }{\left|a\right|({\lambda }_2-{\lambda }_1)}\left(\frac{1}{{\lambda }_1}-\frac{1}{{\lambda }_2}\right)=\frac{\epsilon }{\left|a\right|\left({\lambda }_1{\lambda }_2\right)}.\hfill \end{array}$$

Case 1 (ii): Let ${\lambda }_1,{\lambda }_2$ satisfy (6) with ${\lambda }_1<{\lambda }_2<0$. Let

$$p\left(t\right)=A\left(\varphi \right)e^{{\lambda }_{1}t}+B\left(\varphi \right)e^{{\lambda }_{2}t},$$

which is a solution of (2). Using (7) and recalling that $\left|q\right(t\left)\right|\le \epsilon$, we have

$$\begin{array}{cc}\hfill \underset{t\in I}{sup}|\varphi \left(t\right)-p\left(t\right)|& \le \underset{t\in I}{sup}\frac{\epsilon }{\left|a\right|({\lambda }_2-{\lambda }_1)}{\int }_0^t\left|e^{{\lambda }_2(t-s)}-e^{{\lambda }_1(t-s)}\right|ds\hfill \\ \hfill & \le \frac{\epsilon }{\left|a\right|({\lambda }_2-{\lambda }_1)}\left(\frac{1}{{\lambda }_1}-\frac{1}{{\lambda }_2}\right)=\frac{\epsilon }{\left|a\right|\left({\lambda }_1{\lambda }_2\right)}.\hfill \end{array}$$

Case 1 (iii): Let ${\lambda }_1,{\lambda }_2$ satisfy (6) with ${\lambda }_2<0<{\lambda }_1$. Let

$$p\left(t\right)=A\left(\varphi \right)e^{{\lambda }_{1}t}+B\left(\varphi \right)e^{{\lambda }_{2}t}-\frac{1}{a({\lambda }_2-{\lambda }_1)}{\int }_0^{\infty }q\left(s\right)e^{{\lambda }_1(t-s)}ds,$$

which exists and is well defined because ${\lambda }_1>0$ and is a solution of (2). Using (7) and recalling that $\left|q\right(t\left)\right|\le \epsilon$, we have

$$\begin{array}{cc}\hfill \underset{t\in I}{sup}|\varphi \left(t\right)-p\left(t\right)|& =\frac{1}{\left|a\right|({\lambda }_1-{\lambda }_2)}\underset{t\in I}{sup}\left|{\int }_0^{t}q\left(s\right)e^{{\lambda }_2(t-s)}ds+{\int }_t^{\infty }q\left(s\right)e^{{\lambda }_1(t-s)}ds\right|\hfill \\ \hfill & \le \frac{\epsilon }{\left|a\right|({\lambda }_1-{\lambda }_2)}\underset{t\in I}{sup}\left({\int }_0^t{e}^{{\lambda }_2(t-s)}ds+{\int }_t^{\infty }{e}^{{\lambda }_1(t-s)}ds\right)\hfill \\ \hfill & \le \frac{\epsilon }{\left|a\right|({\lambda }_1-{\lambda }_2)}\left(\frac{1}{|{\lambda }_2|}+\frac{1}{{\lambda }_1}\right)=\frac{\epsilon }{\left|a\right|{\lambda }_1\left|{\lambda }_2\right|}.\hfill \end{array}$$

Case 1 (iv): Let ${\lambda }_1=\alpha -\beta i,{\lambda }_2=\alpha +\beta i\in \mathbb{C}$ satisfy (6) with $\alpha ,\beta >0$. Recall that $sin\theta =\frac{e^{i\theta }-e^{-i\theta }}{2i}$, and let

$$\begin{array}{cc}\hfill p\left(t\right)& =A\left(\varphi \right)e^{{\lambda }_{1}t}+B\left(\varphi \right)e^{{\lambda }_{2}t}+\frac{1}{a({\lambda }_2-{\lambda }_1)}{\int }_0^{\infty }q\left(s\right)\left(e^{{\lambda }_2(t-s)}-e^{{\lambda }_1(t-s)}\right)ds\hfill \\ \hfill & =A\left(\varphi \right)e^{{\lambda }_{1}t}+B\left(\varphi \right)e^{{\lambda }_{2}t}+\frac{1}{a\beta }{\int }_0^{\infty }q\left(s\right)sin\left(\beta (t-s)\right)e^{\alpha (t-s)}ds,\hfill \end{array}$$

which exists and is well defined because $\alpha >0$ and is a solution of (2). Using (7) and noting that $\left|q\right(t\left)\right|\le \epsilon$, we have

$$\begin{array}{cc}\hfill \underset{t\in I}{sup}|\varphi \left(t\right)-p\left(t\right)|& =\underset{t\in I}{sup}\left|\frac{1}{a\beta }{\int }_t^{\infty }q\left(s\right)sin\left(\beta (t-s)\right)e^{\alpha (t-s)}ds\right|\hfill \\ \hfill & =\underset{t\in I}{sup}\left|\frac{1}{a{\beta }^2}{\int }_0^{\infty }q\left(\frac{u}{\beta }+t\right)sin(-u)e^{\frac{-\alpha u}{\beta }}du\right|\hfill \\ \hfill & \le \frac{\epsilon }{\left|a\right|{\beta }^2}{\int }_0^{\infty }|sin\left(u\right)|e^{\frac{-\alpha u}{\beta }}du\hfill \\ \hfill & =\frac{1}{c}coth\left(\frac{-b\pi }{2\sqrt{4ac-b^2}}\right)\hfill \\ \hfill & =\frac{1}{c}coth\left(\frac{\left|b\right|\pi }{2\sqrt{4ac-b^2}}\right).\hfill \end{array}$$

Case 1 (v): Let ${\lambda }_1=\alpha -\beta i,{\lambda }_2=\alpha +\beta i\in \mathbb{C}$ satisfy (6) with $\alpha <0<\beta$. Let

$$p\left(t\right)=A\left(\varphi \right)e^{{\lambda }_{1}t}+B\left(\varphi \right)e^{{\lambda }_{2}t},$$

which exists, is well defined, and is a solution of (2). Using (7) and recalling that $\left|q\right(t\left)\right|\le \epsilon$, we have

$$\begin{array}{cc}\hfill \underset{t\in I}{sup}|\varphi \left(t\right)-p\left(t\right)|& =\underset{t\in I}{sup}\frac{1}{\left|a\right|\beta }\left|{\int }_0^{t}q\left(s\right)sin\left(\beta (t-s)\right)e^{\alpha (t-s)}ds\right|\hfill \\ \hfill & =\underset{t\in I}{sup}\frac{1}{\left|a\right|{\beta }^2}\left|{\int }_0^{\beta t}q\left(t-\frac{u}{\beta }\right)sin\left(u\right)e^{\frac{\alpha u}{\beta }}du\right|\hfill \\ \hfill & \le \frac{1}{\left|a\right|{\beta }^2}{\int }_0^{\infty }|sin\left(u\right)|e^{\frac{\alpha u}{\beta }}du\hfill \\ \hfill & =\frac{1}{c}coth\left(\frac{\left|b\right|\pi }{2\sqrt{4ac-b^2}}\right).\hfill \end{array}$$

Case 1 (vi): Let ${\lambda }_1,{\lambda }_2\in \mathbb{C}$ satisfy (6) with ${\lambda }_1=-i\beta$, ${\lambda }_2=i\beta$. For arbitrary $\epsilon >0$, let $q\left(s\right)=\epsilon cos\left(\beta s\right)$, and let $\varphi$ satisfy (7) with $\varphi \left(0\right)=0={\varphi }^{\prime }\left(0\right)$ and this q. Then

$$\varphi \left(t\right)=\frac{1}{a({\lambda }_2-{\lambda }_1)}{\int }_0^{t}q\left(s\right)\left(e^{{\lambda }_2(t-s)}-e^{{\lambda }_1(t-s)}\right)ds=\frac{\epsilon tsin\left(\beta t\right)}{2a\beta },$$

and since any solution of (2) is $p\left(t\right)=c_{1}cos\left(\beta t\right)+c_{2}sin\left(\beta t\right)$ for constants $c_1,c_2\in \mathbb{R}$, we have

$$\underset{t\in (0,\infty )}{sup}|\varphi \left(t\right)-p\left(t\right)|=\infty ,$$

making (2) unstable.

Case 1 (vii): Let ${\lambda }_1\ne {\lambda }_2\in \mathbb{R}$ satisfy (6) with ${\lambda }_1=0$, ${\lambda }_2=-\frac{b}{a}$; in other words, $b\ne 0$ and $c=0$ in (2). For arbitrary $\epsilon >0$, let $q\left(s\right)\equiv \epsilon$, and let $\varphi$ satisfy (7) with $\varphi \left(0\right)=0={\varphi }^{\prime }\left(0\right)$ and this q. Then,

$$\varphi \left(t\right)=\frac{\epsilon }{b^2}\left[a\left(-1+e^{-\frac{bt}{a}}\right)+bt\right],$$

and since any solution of (2) is $p\left(t\right)=c_1+c_2{e}^{-\frac{bt}{a}}$ for constants $c_1,c_2\in \mathbb{R}$, we have

$$\underset{t\in (0,\infty )}{sup}|\varphi \left(t\right)-p\left(t\right)|=\infty ,$$

making (2) unstable.

Case 2: Let ${\lambda }_1,{\lambda }_2$ satisfy (6) with ${\lambda }_1={\lambda }_2=-\frac{b}{2a}$. Assume $p\left(t\right)$ has a Kamal transform $\mathcal{K}\left\{p\right(t\left)\right\}=P\left(s\right)$. Then,

$$\mathcal{K}\left\{p^{″}\left(t\right)\right\}=\mathcal{K}\left\{{\left(p^{\prime }\right)}^{\prime }\left(t\right)\right\}=\frac{1}{s^2}P\left(s\right)-\frac{1}{s}p\left(0\right)-p^{\prime }\left(0\right),$$

so the Kamal transform of (2) is

$$\frac{a}{s^2}P\left(s\right)-\frac{a}{s}p\left(0\right)-a{p}^{\prime }\left(0\right)+\frac{b}{s}P\left(s\right)-bp\left(0\right)+cP\left(s\right)=0.$$

Solving this for $P\left(s\right)$ yields

$$P\left(s\right)=\frac{\left(\frac{a}{s}+b\right)p\left(0\right)+a{p}^{\prime }\left(0\right)}{\frac{a}{s^2}+\frac{b}{s}+c}=\frac{\frac{p\left(0\right)}{s}+\frac{bp\left(0\right)}{a}+p^{\prime }\left(0\right)}{{\left(\frac{1}{s}-{\lambda }_1\right)}^2}=\frac{A\left(p\right)}{\left(\frac{1}{s}-{\lambda }_1\right)}+\frac{B\left(p\right)}{{\left(\frac{1}{s}-{\lambda }_1\right)}^2},$$

where $0<s<{\lambda }_1^{-1}$ if ${\lambda }_1>0$ for

$$A\left(p\right)=p\left(0\right)\mathrm{and}B\left(p\right)=\left({\lambda }_1+\frac{b}{a}\right)p\left(0\right)+p^{\prime }\left(0\right).$$

Using the inverse Kamal transform on $P\left(s\right)$ gives us the solution

$$p\left(t\right)=A\left(p\right)e^{{\lambda }_{1}t}+B\left(p\right)t{e}^{{\lambda }_{1}t}$$

for (2). Given arbitrary $\epsilon >0$, suppose $\varphi :I\to \mathbb{R}$ satisfies

$$a{\varphi }^{″}\left(t\right)+b{\varphi }^{\prime }\left(t\right)+c\varphi \left(t\right)=q\left(t\right),\mathrm{where}\underset{t\in I}{sup}\left|q\left(t\right)\right|\le \epsilon .$$

Then, the Kamal transform and basic algebra yield

$$\Phi \left(s\right)=\frac{A\left(\varphi \right)}{\left(\frac{1}{s}-{\lambda }_1\right)}+\frac{B\left(\varphi \right)}{{\left(\frac{1}{s}-{\lambda }_1\right)}^2}+\frac{Q\left(s\right)}{a{\left(\frac{1}{s}-{\lambda }_1\right)}^2},$$

so that the inverse Kamal transform gives us

$$\varphi \left(t\right)=A\left(\varphi \right)e^{{\lambda }_{1}t}+B\left(\varphi \right)t{e}^{{\lambda }_{1}t}+\frac{1}{a}{\int }_0^t(t-s)e^{{\lambda }_1(t-s)}q\left(s\right)ds.$$

(8)

Case 2 (i): Let ${\lambda }_1,{\lambda }_2$ satisfy (6) with ${\lambda }_2={\lambda }_1>0$. Let

$$p\left(t\right)=A\left(\varphi \right)e^{{\lambda }_{1}t}+B\left(\varphi \right)t{e}^{{\lambda }_{1}t}+\frac{1}{a}{\int }_0^{\infty }(t-s)e^{{\lambda }_1(t-s)}q\left(s\right)ds,$$

which exists and is well defined because ${\lambda }_2={\lambda }_1>0$ and is a solution of (2). Using (7) and recalling that $\left|q\right(t\left)\right|\le \epsilon$, we have

$$\begin{array}{cc}\hfill \underset{t\in I}{sup}|\varphi \left(t\right)-p\left(t\right)|& =\underset{t\in I}{sup}\frac{1}{\left|a\right|}\left|-{\int }_t^{\infty }(t-s)e^{{\lambda }_1(t-s)}q\left(s\right)ds\right|\hfill \\ \hfill & \le \underset{t\in I}{sup}\frac{\epsilon }{\left|a\right|}{\int }_t^{\infty }\left|(t-s)e^{{\lambda }_1(t-s)}\right|ds=\frac{\epsilon }{|a{\lambda }_1^2|}.\hfill \end{array}$$

Case 2 (ii): Let ${\lambda }_1,{\lambda }_2$ satisfy (6) with ${\lambda }_1={\lambda }_2<0$. Let

$$p\left(t\right)=A\left(\varphi \right)e^{{\lambda }_{1}t}+B\left(\varphi \right)t{e}^{{\lambda }_{1}t},$$

which is a solution of (2). Using (8) and recalling that $\left|q\right(t\left)\right|\le \epsilon$, we have

$$\begin{array}{cc}\hfill \underset{t\in I}{sup}|\varphi \left(t\right)-p\left(t\right)|& \le \underset{t\in I}{sup}\frac{\epsilon }{\left|a\right|}{\int }_0^t\left|(t-s)e^{{\lambda }_1(t-s)}\right|ds\hfill \\ \hfill & =\underset{t\in I}{sup}\frac{\epsilon }{\left|a\right|}\left(\frac{1+e^{{\lambda }_{1}t}({\lambda }_{1}t-1)}{{\lambda }_1^2}\right)=\frac{\epsilon }{|a{\lambda }_1^2|}.\hfill \end{array}$$

Case 2 (iii): Let ${\lambda }_1,{\lambda }_2$ satisfy (6) with ${\lambda }_1={\lambda }_2=0$. Let

$$p\left(t\right)=A\left(p\right)+B\left(p\right)t,$$

which is a solution of (2). For arbitrary $\epsilon >0$, let $q\left(s\right)\equiv \epsilon$, and let $\varphi$ satisfy (8) with $\varphi \left(0\right)=0={\varphi }^{\prime }\left(0\right)$ and this q. Then, $\varphi \left(t\right)=\frac{\epsilon t^2}{2a}$, and

$$\underset{t\in I}{sup}|\varphi \left(t\right)-p\left(t\right)|\le \underset{t\in I}{sup}\left|\frac{\epsilon t^2}{2a}-A\left(p\right)-B\left(p\right)t\right|=\infty$$

for any choice of $p\left(0\right)$ and $p^{\prime }\left(0\right)$. Thus, (2) is unstable in this case. □

Remark 1. 

The results in Theorems 1 and 2 improve those found in [4] using the Kamal transform. Moreover, the techniques used in this paper can be modified to obtain similar Hyers–Ulam stability results for linear constant coefficient equations of first and second order using the Laplace, Tarig, Aboodh, Mahgoub, Sawi, Shehu, and Elzaki integral transforms, respectively. For example, by including information on the best Hyers–Ulam constant, Theorem 1 slightly improves [1] (Theorem 3.3) and [8] (Theorem 3.3), which used the Laplace transform and the Maghoub transform, respectively. Moreover, Theorem 2 improves [1] (Theorem 3.4) and [8] (Theorem 3.4) for the second-order case.

The results in Theorem 2 also improve those found in [24,25], which only consider (2) on a finite interval. Theorem 2 is also different from the results in [12] (Section 3), using the Aboodh transform. Theorem 2 matches the results found in [26] for the case where the characteristic roots satisfy ${\lambda }_1,{\lambda }_2\in \mathbb{R}$ using a different method and extends those results to include the case where ${\lambda }_1,{\lambda }_2\in \mathbb{C}$.

3. Examples

In this section, several key examples are provided to illustrate the applicability of our results for the second-order Equation (2). We also compare and contrast our results with those in the extant literature.

Example 1. 

Equation (2) in the form

$$p^{″}\left(t\right)\pm 2\eta p^{\prime }\left(t\right)+cp\left(t\right)=0,c\in \mathbb{R},\eta ,t\in (0,\infty )$$

(9)

is Hyers–Ulam-stable if and only if $c\ne 0$, with the Hyers–Ulam stability constant

$$K=\left\{\begin{array}{cc}\frac{1}{\left|c\right|}\hfill & ifc\in (-\infty ,0)\cup \left(0,{\eta }^2\right],\hfill \\ \frac{1}{c}coth\left(\frac{\eta \pi }{2\sqrt{c-{\eta }^2}}\right)\hfill & ifc\in \left({\eta }^2,\infty \right).\hfill \end{array}\right.$$

Proof. 

The roots of the corresponding characteristic equation are $-\eta \pm \sqrt{{\eta }^2-c}$ for $b=2\eta >0$, and $\eta \pm \sqrt{{\eta }^2-c}$ for $b=-2\eta <0$, respectively. This proof details the Hyers–Ulam stability of (9) for all possible values of $c\in \mathbb{R}$ and $\eta >0$ for $b=2\eta$. The proof for $b=-2\eta$ is similar and thus omitted.

(a) If $c\in (-\infty ,0)$, then $-\eta -\sqrt{{\eta }^2-c}<0<-\eta +\sqrt{{\eta }^2-c}$. From the proof of Theorem 2 Case 1 (iii), we have the result that (9) is Hyers–Ulam-stable with the Hyers–Ulam stability constant

$$K=\frac{1}{\left(-\eta +\sqrt{{\eta }^2-c}\right)\left|-\eta -\sqrt{{\eta }^2-c}\right|}=\frac{1}{-c}=\frac{1}{\left|c\right|}.$$

(b) If $c=0$, then ${\lambda }_1=0$ and ${\lambda }_2=-2\eta$. From the proof of Theorem 2 Case 1 (vii), we have the result that (9) is not Hyers–Ulam-stable.

(c) If $c\in (0,{\eta }^2)$, then $-\eta -\sqrt{{\eta }^2-c}<-\eta +\sqrt{{\eta }^2-c}<0$. From the proof of Theorem 2 Case 1 (ii), we have the result that (9) is Hyers–Ulam-stable with the Hyers–Ulam stability constant

$$K=\frac{1}{(-\eta +\sqrt{{\eta }^2-c})(-\eta -\sqrt{{\eta }^2-c})}=\frac{1}{c}.$$

(d) If $c={\eta }^2$, then ${\lambda }_1={\lambda }_2=-\eta <0$. From the proof of Theorem 2 Case 2 (ii), we have the result that (9) is Hyers–Ulam-stable with the Hyers–Ulam stability constant

$$K=\frac{1}{{\lambda }_1^2}=\frac{1}{{\eta }^2}=\frac{1}{c}.$$

(e) If $c\in \left({\eta }^2,\infty \right)$, then the characteristic roots are $\alpha \mp \beta i=-\eta \mp i\sqrt{c-{\eta }^2}$ with $\alpha =-\eta <0$. From the proof of Theorem 2 Case 1 (v), we have the result that (9) is Hyers–Ulam-stable with the Hyers–Ulam stability constant

$$K=\frac{1}{c}coth\left(\frac{\eta \pi }{2\sqrt{c-{\eta }^2}}\right).$$

This ends the proof of this example. □

Remark 2. 

The authors in [12] (Example 5.2) consider the equation

$$p^{″}\left(t\right)-4p^{\prime }\left(t\right)+3p\left(t\right)=sint$$

and state that the Hyers–Ulam constant is $\frac{1}{12}$. However, according to [26] (Theorem 4.1 (i)), the best (minimal) constant is $\frac{1}{3}$. Note that in (9) above with $\eta =2$ and $c=3$ and a homogeneous right-hand side, the Hyers–Ulam constant in this example is $K=\frac{1}{c}=\frac{1}{3}$ as well. Moreover, using different methods, the authors in [22] (Theorem 5) prove that (2) with $a=1$ and $b,c\in \mathbb{R}\setminus \left\{0\right\}$ with $4ac-b^2>0$ has best constant

$$K=\frac{1}{c}coth\left(\frac{\left|b\right|\pi }{2\sqrt{4c-b^2}}\right),$$

which matches the results in Theorem 2 and this example. In the next example, we show that the assumptions $a=1$ and $b,c\in \mathbb{R}\setminus \left\{0\right\}$ can be weakened; that is, we let $a\in \mathbb{R}\setminus \left\{0\right\}$ and $b=0$ with $c\in \mathbb{R}$, to obtain Hyers–Ulam stability for (2) if and only if $\frac{c}{a}<0$.

Example 2. 

Equation (2) in the form

$$a{p}^{″}\left(t\right)+cp\left(t\right)=0,a\in \mathbb{R}\setminus \left\{0\right\},c\in \mathbb{R},t\in (0,\infty )$$

(10)

is Hyers–Ulam-stable if and only if $\frac{c}{a}<0$, with the Hyers–Ulam stability constant $K=\frac{1}{\left|c\right|}$.

Proof. 

The roots of the corresponding characteristic equation are $\pm \sqrt{-\frac{c}{a}}$.

(a) If $\frac{c}{a}\in (-\infty ,0)$, then $-\sqrt{-\frac{c}{a}}<0<\sqrt{-\frac{c}{a}}$. From the proof of Theorem 2 Case 1 (iii), we have the result that (10) is Hyers–Ulam-stable with the Hyers–Ulam stability constant

$$K=\frac{1}{\left(\sqrt{-\frac{c}{a}}\right)\left|-\sqrt{-\frac{c}{a}}\right|}=\frac{1}{\left|c\right|}.$$

(b) If $c=0$, then ${\lambda }_1=0={\lambda }_2$. From the proof of Theorem 2 Case 2 (iii), we have the result that (10) is not Hyers–Ulam-stable.

(c) If $\frac{c}{a}\in (0,\infty )$, then the characteristic roots are $\alpha \mp \beta i=\mp i\sqrt{\frac{c}{a}}$ with $\alpha =0$. From the proof of Theorem 2 Case 1 (vi), we have the result that (10) is not Hyers–Ulam-stable. □

Example 3. 

Equation (2) in the form

$$p^{″}\left(t\right)+b{p}^{\prime }\left(t\right)+p\left(t\right)=0,b\in \mathbb{R},t\in (0,\infty )$$

(11)

is Hyers–Ulam-stable if and only if $b\ne 0$, with the Hyers–Ulam stability constant

$$K=\left\{\begin{array}{cc}1\hfill & ifb\in (-\infty ,-2]\cup [2,\infty ),\hfill \\ coth\left(\frac{\left|b\right|\pi }{2\sqrt{4-b^2}}\right)\hfill & ifb\in (-2,0)\cup (0,2).\hfill \end{array}\right.$$

Proof. 

The roots of the corresponding characteristic equation are $\frac{-b\pm \sqrt{b^2-4}}{2}$. The proof below explores the Hyers–Ulam stability of (11) for all possible values of $b\in \mathbb{R}$.

(a) If $b\in (-\infty ,-2)$, then $\frac{-b\pm \sqrt{b^2-4}}{2}>0$. From the proof of Theorem 2 Case 1 (i), we have the result that (11) is Hyers–Ulam-stable with the Hyers–Ulam stability constant

$$K=\frac{1}{{\lambda }_1{\lambda }_2}=\frac{1}{1}=1.$$

(b) If $b=-2$, then ${\lambda }_1={\lambda }_2=1$. From the proof of Theorem 2 Case 2 (i), we have the result that (11) is Hyers–Ulam-stable with the Hyers–Ulam stability constant

$$K=\frac{1}{{\lambda }_1^2}=1.$$

(c) If $b\in (-2,0)$, then $\alpha \mp \beta i=\frac{-b\mp i\sqrt{4-b^2}}{2}$ with $\alpha >0$. From the proof of Theorem 2 Case 1 (iv), we have the result that (11) is Hyers–Ulam-stable with the Hyers–Ulam stability constant

$$K=coth\left(\frac{\left|b\right|\pi }{2\sqrt{4-b^2}}\right).$$

(d) If $b=0$, then the characteristic roots are $\alpha \mp \beta i=\mp i$ with $\alpha =0$. From the proof of Theorem 2 Case 1 (vi), we have the result that (11) is not Hyers–Ulam-stable.

(e) If $b\in (0,2)$, then $\alpha \mp \beta i=\frac{-b\mp i\sqrt{4-b^2}}{2}$ with $\alpha <0$. From the proof of Theorem 2 Case 1 (v), we have the result that (11) is Hyers–Ulam-stable with the Hyers–Ulam stability constant

$$K=coth\left(\frac{b\pi }{2\sqrt{4-b^2}}\right).$$

(f) If $b=2$, then ${\lambda }_1={\lambda }_2=-1$. From the proof of Theorem 2 Case 2 (ii), we have the result that (11) is Hyers–Ulam-stable with the Hyers–Ulam stability constant

$$K=\frac{1}{{\lambda }_1^2}=1.$$

(g) If $b\in (2,\infty )$, then $\frac{-b\pm \sqrt{b^2-4}}{2}<0$. From the proof of Theorem 2 Case 1 (ii, we have the result that (11) is Hyers–Ulam-stable with the Hyers–Ulam stability constant

$$K=\frac{1}{{\lambda }_1{\lambda }_2}=\frac{1}{1}=1.$$

This ends the proof of this example. □

Remark 3. 

In recent papers [9] (Section 4 Theorem 3) using the Tarig transform, [10] (Theorem 4.1) using the Shehu transform, [11] (Theorem 5.1) using the Sawi transform, [20] (Theorem 4.1) using the Elzaki transform, it is claimed for equations of order two that if $(b+c)$ is constant and $Re(b+c)>0$, then (2) with $a=1$, namely

$$p^{″}\left(t\right)+b{p}^{\prime }\left(t\right)+cp\left(t\right)=0,b,c\in \mathbb{R},t\in (0,\infty ),$$

is Hyers–Ulam-stable. Let $c=1$ and consider Example 3 above. If $b=-4$, then $Re(b+c)=-3<0$ and the equation is Hyers–Ulam-stable. If $b=0$, then $Re(b+c)=1>0$, but the equation is not Hyers–Ulam-stable. Thus, the results in this paper are different from those in [9,10,11,20].

Remark 4. 

Another recent paper [27] considered (2) with $a=1$, $b=\gamma -1$, and $c=-\gamma$ for any $\gamma \in \mathbb{R}$, namely

$$p^{″}\left(t\right)+(\gamma -1)p^{\prime }\left(t\right)-\gamma p\left(t\right)=0,\gamma \in \mathbb{R},t\in [A,B].$$

Note that this is on the finite interval $-\infty <A<t<B<\infty$ and that in [27] (Theorem 3.2) the authors ignored the case of $\gamma =0$. The following example extends the interval to $(0,\infty )$ and considers all values of $\gamma \in \mathbb{R}$.

Example 4. 

Equation (2) in the form

$$p^{\prime \prime }\left(t\right)+(\gamma -1)p^{\prime }\left(t\right)-\gamma p\left(t\right)=0,\gamma \in \mathbb{R},t\in (0,\infty )$$

(12)

is Hyers–Ulam-stable if and only if $\gamma \ne 0$, with the Hyers–Ulam stability constant

$$K=\frac{1}{\left|\gamma \right|}.$$

Proof. 

The roots of the corresponding characteristic equation are $\frac{(1-\gamma )\pm |\gamma +1|}{2}$, respectively. The proof below explores the Hyers–Ulam stability of (12) for all possible values of $\gamma \in \mathbb{R}$.

(a) If $\gamma \in (-\infty ,-1)\cup (-1,0)$, then $\lambda =-\gamma ,1>0$. From the proof of Theorem 2 Case 1 (i), we have the result that (12) is Hyers–Ulam-stable with the Hyers–Ulam stability constant

$$K=\frac{1}{{\lambda }_1{\lambda }_2}=\frac{1}{-\gamma }=\frac{1}{\left|\gamma \right|}.$$

(b) If $\gamma =-1$, then ${\lambda }_1={\lambda }_2=1$. From the proof of Theorem 2 Case 2 (i), we have the result that (12) is Hyers–Ulam-stable with the Hyers–Ulam stability constant

$$K=\frac{1}{{\lambda }_1^2}=\frac{1}{\left|\gamma \right|}.$$

(c) If $\gamma =0$, then the characteristic roots are $\lambda =0,1$. From the proof of Theorem 2 Case 1 (vii), we have the result that (12) is not Hyers–Ulam-stable.

(d) If $\gamma \in (0,\infty )$, then ${\lambda }_2=-\gamma <0<1={\lambda }_1$. From the proof of Theorem 2 Case 1 (iii), we have the result that (12) is Hyers–Ulam-stable with the Hyers–Ulam stability constant

$$K=\frac{1}{{\lambda }_1\left|{\lambda }_2\right|}=\frac{1}{\gamma }.$$

This ends the proof of this example. □

4. Conclusions

In this work, we employed the Kamal transform to determine the Hyers–Ulam stability (HUS) and Hyers–Ulam stability constants for first-order complex constant coefficient differential equations and for second-order real constant coefficient differential equations, improving on previous results obtained by using the Kamal transform and matching or improving on results found using other methods. In a section of examples, we compare and contrast our results favorably with those established in the literature using means specifically other than the Kamal transform. In several key cases, we improve on the results found in the literature and match sharp results in some cases involving the best (minimal) HUS constant. We also point out explicitly important cases of instability. A future direction may be to apply integral transform methods to a matrix-vector equation such as ${\overrightarrow{p}}^{\prime }\left(t\right)+M\overrightarrow{p}\left(t\right)=\overrightarrow{0}$ for a square constant matrix M.