Laplace Transform, Non-Constant Coefficients Differential Equations and Applications to Riccati Equation

In this paper, the Laplace Transform is used to find explicit solutions of a family of second order Differential Equations with non-constant coefficients. For some of these equations, it is possible to find the solutions using standard techniques of solving Ordinary Differential Equations. For others, it seems to be very difficult indeed impossible to find explicit solutions using traditional methods. The Laplace transform could be an alternative way. An application on solving a Riccati Equation is given. Recall that the Riccati Equation is a non-linear differential equation that arises in many topics of Quantum Mechanics and Physics.


Introduction
The concept of Laplace Transform has been intensively used in diverse areas of Science and Engineering, for instance in electric circuit analysis, in communication engineering [1] [2] [3] [4].
It is also a powerful mathematical tool to solve non-homogeneous constant coefficients linear differential equations, especially when the forcing represents a discontinuous function as the Heaviside function or Dirac function [5] [6].
Like many operators, the Laplace Transform has the ability to change any ordinary linear differential equations with constant coefficients into algebraic equations.
It has been already used to find the explicit solutions of some non-constant coefficients linear differential equations. In [7], the authors used a new version of the Laplace Transform called the Sumudu Transform to find the explicit solution of the following non-constant coefficients differential equations along with their initial conditions: A similar work was performed in [8] and [9], where the authors used a different version of the Laplace Transform, this time called the Elkazi Transform, to find the explicit solutions of the initial value problems: It would be very difficult, indeed impossible, to find explicit solutions to some of these types of initial value problems using standard methods. In this paper, the Laplace Transform is used to solve analytically a family of non-constant coefficients second order linear differential equations. In general, non-constant coefficients differential equations are still very difficult to be solved analytically. All the initial value problems listed above are particular cases of the family of non-constant linear differential equations found in this paper.
The paper is divided into five sections. In Section 2, the concepts, properties and the existence of the Laplace transform are introduced.
In Section 3, conditions under which a family of non-constant coefficients ordinary differential equations can be solved quantitatively by using the Laplace Transform will be discussed. Specific examples of non-constant coefficients differential equations that satisfy those conditions will be given. Section 4 gives an application to the Riccati Equation.
Section 5 is dedicated to the conclusion.

Definition, Existence and Properties of Laplace Transform
for all numbers s for which this improper integral converges.
The existence of the Laplace Transform of a given function has been discussed in [1] [2].
The following theorem gives a large class of functions for which the Laplace Transform exists, that is the improper integral converges. above theorem are said to be of exponential order a.
Next are examples of functions of exponential order. Example 1.
( ) e bt f t = , ( ) ( ) Remark 2. As a practical matter, most of the functions encountered in the applications are of exponential order.
Theorem 2. If f is of exponential order then ( ) n f is of exponential order for all n.
Proof. We just have to show that if f is of exponential order then f ′ is of exponential order. Suppose that f ′ is of exponential order.
According to the Weierstrass Approximation theorem, any continuous function can be approximated as closely as desired by a polynomial function. That is there exists a sequence of polynomials ( ) This property means that if f is continuous on [ ) 0, ∞ and if ( ) M is called the inverse of  and  is also linear.

Explicit Solutions for a Family of Non-Constant Coefficients Linear Second Order Differential Equations
The goal in this section is to solve analytically non-constant coefficients linear second order differential equations: q , 1 q and 2 q will be stated so that the second order differential equations will have explicit solutions, that is expressed in term of elementary functions.
Using integration by part and the fact that y is of exponential order, we obtain We can use (1), the fact that y′ is of exponential order and Theorem 2 to find (2).
First, we have the following , we obtain (2). That is Under the following conditions: Therefore the explicit solution of (E) is given by: Proof. The first step is to take the Laplace Transform of both sides of (E).
Using the linearity of the Laplace Transform, we get: Using (3), (4) and (5) in (8), and after simplification, we obtain the following equation: We deal with the family of non-constant coefficients linear ordinary differential equations: Using (6) and we get the following equation: Using the same procedure as in example 2, The explicit solution can be written as:  s is given by (7). Therefore the solution to (E) is given by: Proof. The proof is straightforward and is done the same way as in theorem 3. Notice that this solution could be easily found by changing Equation (12) into this following linear equation: q t y c q t y c q t y g t t t + + = So using a traditional method, we can find exactly the same result. Now focusing on the example listed in the introduction, we can show that they all can be obtained using the conditions described in the theorem 4 or theorem 5 given later.

Application to the Riccati Equation
The Riccati Equation named after the mathematician Jacopo Francesco Riccati [10] is the simplest non-linear differential equation. It can be written as: Riccati Equation naturally arises in many fields. Many equations in Physics, Cosmology and Quantum mechanics involve or can be changed into a Riccati Equation [11]. It is well known that there is no general way to solve the Riccati equation. When a particular solution 1 y is found for the Riccati equation, we can use the change 1 u y y = − to find the general solution of the Riccati Equa-very fortunate to find cases in which the inverse of the Laplace Transform can be found easily and therefore, an explicit solution follows. Now for further research, we can try to figure out what change of variable would turn the Riccati Equation into a non-homogeneous non-constant coefficients second order differential equation. This would lead to more applications in solving the Riccati equation using the results from Section 2.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.