Riccati Equation

The Riccati equation is one of the most interesting nonlinear differential equations of first order. It's written in the form:

where a (x), b (x), c (x) are continuous functions of x.
general Riccati equation can be solved with help of the following theorem:
Theorem: If a particular solution y ₁ of a Riccati equation is known, the general solution of the equation is given by

Indeed, substituting the solution y = y ₁ + u into Riccati equation, we have

As a result we obtain the differential equation for the function u (x):

which is a Bernoulli equation. Substitution of z = 1/ u converts the given Bernoulli equation into a linear differential equation that allows integration.
Case 1: Coefficients a, b, c are constants.

If the coefficients in the Riccati equation are constants, this equation can be reduced to a separable differential equation. The solution is described by the integral of a rational function with a quadratic function in the denominator:

Case 2: Equation of type y' = by ² + cx ⁿ

Consider a Riccati equation of type y' = by ² + cxⁿ, where the function a (x) at the linear term is zero, the coefficient b at y ² is a constant, and c (x) is a power function:

Solve the differential equation y' = y + y ² + 1. Solution. The given equation is a simple Riccati equation with constant coefficients. Here the variables x, y can be easily separated, so the general solution of the equation is given by Bernoulli equation is one of the well known nonlinear differential equations of the first order. It is written as where a (x) and b (x) are continuous functions. If m = 0, the equation becomes a linear differential equation. In case of m = 1, the equation becomes separable. In general case, when m ≠ 0, 1, Bernoulli equation can be converted to a linear differential equation using the change of variable The new differential equation for the function z (x) has the form: and can be solved by the methods described on the page Linear Differential Equation of First Order. Find the general solution of the equation y' − y = y²e^x. Solution. We set m = 2 for the given Bernoulli equation, so we use the substitution Differentiating both sides of the equation (we consider y in the right side as a composite function of x), we obtain: Divide both sides of the original differential equation by y ²: Substituting z and z', we find We get the linear equation for the function z (x). To solve it, we use the integrating factor: Then the general solution of the linear equation is given by Returning to the function y (x), we obtain the implicit expression: which can be written in the form: Note that we have lost the solution y = 0 when dividing the equation by y ². Thus, the final answer is given by

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