An Analytical Method for Solving General Riccati Equation

In this paper, the general Riccati equation is analytically solved by a new transformation. By the method developed, looking at the transformed equation, whether or not an explicit solution can be obtained is readily determined. Since the present method does not require a proper solution for the general solution, it is especially suitable for equations whose proper solutions cannot be seen at first glance. Since the transformed second order linear equation obtained by the present transformation has the simplest form that it can have, it is immediately seen whether or not the original equation can be solved analytically. The present method is exemplified by several examples.


I. INTRODUCTION HE generalized Riccati equation is defined by
Here, , , and are arbitrary functions of . This equation is widely encountered in analytical mechanics, engineering and other fields. Therefore, depending on the functions , and , several methods have been developed to solve various types of Riccati equations [3], [4].
If 0, then the transformation 1 ⁄ reduces (1) into a first order linear equation. For the case of 0, several methods are available in the literature. The equation in this case is also known as Bernoulli equation.
The classical method for solving Riccati equation makes the transformation 1 ⁄ , where is a known solution of the equation. However, it is not possible to see a proper solution to the equation at every time. Therefore, this method has a limited usage. Among the other methods, Rao transformation and Allen-Stein transformation can be mentioned. The basic idea here is to bring the main equation into a separable form [1], [7], [8]. However, the differentiability condition is a serious problem with these methods.
Harko et al. have investigated the Riccati Equation and developed a restricted analytical solution [2]. Since the method requires specific conditions among the coefficients in the equation, it cannot be considered a general method. Sugai transformed Riccati equation into a second order differential equation by suggesting a new transformation [11]. Since the transformed equation is more complicated and unsolvable in most cases, it is also not general and applicable. Rao and Ukidave reduced Riccati equation into a separable form under restricted condition [9]. It is of no importance in respect of engineering application. Siller also investigated a separability condition of the equation [10].
Integrability condition for the Riccati equation has been studied by Mak and Harko, and a new method to generate analytical solutions of the Riccati equation was presented [5]. Mortici gives a new method of the variation of constants which leads directly to an equation with separable variables. This method also imposes several restrictions on the solution. Therefore, it cannot also be considered a general method [6].
When all methods are investigated, we observe that no method explains the question about whether or not the analytical solution is obtained in explicit, implicit, or power series form. In addition to finding the analytical solution to the problem for arbitrary values of , , and , the present method answers these important questions.

II. SPECIAL TYPE RICCATI EQUATION
In order to solve (1), consider a new transformation in the form (2) where , are functions to be determined in a convenient manner. We differentiate both sides of (2) twice: and Equation (4)  then, by (5), in the first place, we can have The functions and are to be obtained such that (6) are satisfied. Now, using the inverse transformation from (2), we obtain ln (9) We recall that, for some simple types of Riccati equations, the solution is obtained without having any difficulty in solving the second order equation ( (8)). In the other words, the solution is found without solving a complex second order differential equation with variable coefficients. The examples that are numbered from 1 to 4 illustrate the method.
Example 1: We first try to solve the equation whose solution can also be found by integration. Comparing (10) with (7) gives Inserting 1 into (11a) and solving the equation yields (12) Equation (11c) is automatically satisfied. Now, using (9), we have ln ln (13) It can be verified that (14) satisfies (10). Example 2: We now solve the equation Comparison of (15) with (7) yields The function can be obtained via (16a) and (16b) Using (9) 20) is required to be solved.
Comparison of (20) with (7) yields The function can readily be obtained Using (9), we obtain ln ln . .
In the examples ever seen, the examples satisfying the condition ′′ ⁄ automatically have been chosen. Otherwise, the unknown function must be found such that the equations must be simultaneously satisfied.
However, it is not always possible to obtain a unique that satisfies both (25a) and (25c). To show this case, we The function can be obtained as We then have (29) Thus, the present method gives a solution as long as the condition given in (29) In order to remove the restriction in Section II, we now assume that we seek the solution of the equation (37) where is a function to be determined. Thus, (36) gives However, by (33), we can also write Replacing by , we obtain 0 Equation (40) is a linear second order ordinary differential equation whose general solution is generally obtained in power series. However, if is a constant or something that leads to the analytical solution of (40), then the explicit form of or can always be obtained. Now, an attention should be paid to (40). According to this result, we are led to conclude that one of the main advantage of the transformation given by (33) is that it converts Riccati equation directly into the linear form of (40). Equation (40) has the simplest form of a second order differential equation. This form also provides us the knowledge of whether we can obtain the explicit solution of (37) or not. After (40) is solved via classical methods, the inverse transformation yields the solution as ln Since (41) does not involve any expression to be integrated, there exists no difficulty in obtaining the explicit form of . Example 5: As a first example in this section, we solve the equation Comparing (42)  Note that if we take as the solution of , then we obtain the proper solution 1 . In addition, if we take only, then we find the other proper solution of (44) as 1 . Indeed, we can verify that these are proper solutions of (45). Thus, the method presented also gives the proper solutions of Riccati Equation. Then, when desired, one can use the famous transformation ̅ 1 ⁄ to find the general solution. Here, ̅ is a proper solution. Example 7: We now try to solve the equation We recall that a proper solution of (54) is not readily seen. Therefore, the well-known classical method cannot be used. It is obvious that (55a) and (55b) are valid: Solving (55a)  the form of , we can immediately see whether the solution is expressed in explicit or implicit form. Since we do not need any proper solution already required for the analytical solution, the present method is further simple and can be preferred in teaching the solutions of Riccati differential equation. Since the method does not put any condition on the functions involved in the equation, the method is very general.
According to the present method, proper solutions of Riccati equation can also be obtained. Thus, using the conventional method of transforming the original equation into a linear first order differential equation can be followed. This situation is demonstrated in Example 6.