New Conditions for Obtaining the Exact Solutions of the General Riccati Equation
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.
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 : where and are arbitrary differentiable functions in with .
A new integrability condition for the Riccati equation was presented in  under the following constraint: where is an arbitrary differentiable function on .
Also, the general solution was found in  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).
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
3.3. Equations Containing Power Functions
Example 3. Consider the following equation:
where , and .
The condition equation (25) holds if and only if with . So the general solution to this equation can be found for any values of , and by
Example 4. For the equation where , and , we get , and the general solution to this equation is given by
3.4. Other Equations
Example 5. Consider the following Riccati equation: where , and . We get , and the general solution to this equation is given by Thus the exact closed form solution is where is a constant of integration.
Example 6. For the given equation where , and , we get , and the general solution to this equation is given by Thus the exact closed form solution is where is a constant of integration.
3.5. Equations Containing Exponential Functions
Example 7. Consider the following Riccati equation:
where , and .
The condition equation (42) holds if and only if .
Combining this value of with (35), we get . So . Thus the general solution to this equation can be found by (41); that is,
4.1. The Central Potential Problem of the Power Law Type
Certain types of Newtons laws of motion are equivalent to the Riccati equation. For example, the equation for the energy conservation in the case of a central potential is given by the standard expression  Under the influence of a power law central potential and , where is the coupling constant and the exponent can be either positive or negative, (59) can be transformed into the Riccati equation A solution of this equation will lead to a solution of the original equation, where and .
4.2. Damped Harmonic Oscillators
Various problems in quantum optics, superconductivity, and nonrelativistic quantum mechanics can be described classically by  where is the particle coordinate, is the mass of the particle, is a damping constant, and is a potential energy that accounts for the interaction of the particle with its environment. If , where is white or colored noise, then (60) becomes the well-known Langevin equation Also the equation of motion is where is the damping function and is the the natural frequency of the undamped oscillator.
The substitution converts (66) to the Riccati equation Here , and .
The condition (25) holds if and only if the coefficients and satisfy the following condition: Then (68) can be written in an equivalent form as which is a Bernoulli equation and can be readily solved for to obtain We conclude that if the coefficients and satisfy (70), then the general solution of (67) is given by (26).
Returning to the original dependent variable by , we obtain the general solution to (66) as where is a constant.
We have converted the Riccati equation into an equivalent equation; then, by using the integrability condition for this equation, we obtain a separable equation. The first case , where constant is a constant, is considered. Thus, the general solutions of the Riccati equation can be exactly obtained. The second integrability case is obtained for the reduced Riccati equation with an arbitrary function .
We have considered several distinct examples to illustrate our new approach. The method is also applied to the Riccati equation arising in the solution of certain types of Newtons laws of motion and the damped harmonic oscillators equations.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
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