Abstract

We propose a direct method for solving the general Riccati equation . We first reduce it into an equivalent equation, and then we formulate the relations between the coefficients functions , and of the equation to obtain an equivalent separable equation from which the previous equation can be solved in closed form. Several examples are presented to demonstrate the efficiency of this method.

1. Introduction

In realistic investigations of the dynamics of a physical system, nonlinearity may often be in the form of the well-known Riccati equation: The Riccati equation is used in different areas of physics, engineering, and mathematics such as quantum mechanics, thermodynamics, and control theory. It also appears in many engineering design simulations.

The first well-known result in the analysis of the Riccati equation is that [1, 2] if one particular solution can be found to (1), then the general solution is obtained as where satisfies the corresponding Bernoulli equation The substitution that is needed to solve this Bernoulli equation is A set of solutions to the Riccati equation is then given by The second important result in the analysis of the solution of the Riccati equation is that the general equation can always be reduced to a second-order linear differential equation of the form [3, pages 23–25] where A solution of this equation will lead to a solution of the original Riccati equation.

To our knowledge it is often very difficult, if not impossible, to find closed form solutions of such nonlinear differential equations. But a number of solutions of the Riccati equation can be obtained by assuming that the coefficients , and satisfy certain constraints. Indeed, closed-form solutions are known if the following condition is satisfied [4]: where and are arbitrary differentiable functions in with .

A new integrability condition for the Riccati equation was presented in [5] under the following constraint: where is an arbitrary differentiable function on .

Also, the general solution was found in [6] when holds, where and are either constants or and is an arbitrary function.

In this paper, we present new solutions of the general Riccati equation. We first reduce it to an equivalent equation, and then we formulate the relations between the coefficients functions , and of the equation to obtain the required relation where is an arbitrary function. Therefore the given Riccati equation can be transformed into a separable equation, which can be easily solved in two cases: equals a constant , or certain functions. Several examples are studied in detail to illustrate the proposed technique.

2. Exact Solutions of the Riccati Equation

We begin our approach by converting the general Riccati equation (1) to a simpler form by the substitution which yields the equation where and .

Now, let and consider the case . Thus the transformation of (15) to (14) yields Then (16) can be written in an equivalent form as An important remark can be made here. Equation (16) can be solved by assuming that the functions and satisfy the following condition: where is an arbitrary function.

This condition can be written in an equivalent form as or equivalently This leads to the following important cases that solve (16).

2.1. Case 1: Equals a Constant

Substituting into (19) gives which is a first-order separable differential equation, and we can obtain its closed form solution from Based on the integral involving the rational algebraic functions of the form in view of this, the solution in a closed form is given by Once is found then we can obtain from (15).

Finally, after finding , we can use to return to the original variable.

Our results can thus be summarized by the following lemma.

Lemma 1. If the coefficients of the general Riccati equation (1) satisfy the condition where is a constant, then the general solutions of the Riccati equation can be exactly obtained as where is a constant.

Remark 2. For the case , proceeding as before, we obtain the following condition: and the first-order separable equation Thus the general solutions can be similarly found.

2.2. Case 2: Equals an Arbitrary Function

We have This equation can be written in an equivalent form as The substitution of into (30) yields If we assume that the function satisfies the following condition: where is a constant, then which is a separable equation. Thus where is an arbitrary integration constant. Then it is easy to solve (32) in closed form. Thus or where is an arbitrary integration constant. Therefore, Substituting the original expression for , we obtain the final general solution as For , we have Thus

We have the following.

Lemma 3. If the coefficients of the general Riccati equation (1) satisfy the condition where is a function given by (35), then the general closed form solutions of the Riccati equation can be exactly obtained by (39) and (41).

3. Examples

Listed below are some special cases where the above conditions are satisfied and the general solutions of the Riccati equation are found.

3.1. The Coefficients and Are Proportional

Example 1. Consider the following Riccati equation: where , and are proportional; that is, , and .
The condition equation (25) holds if and only if . So the general solution to this equation can be found by

3.2. Special Case Where the Original Equation Has the Canonical Form

Example 2. Consider the following Riccati equation: where , and .
The condition equation (25) holds if and only if . Thus, the general solution is given as