3. Bernoulli Equation. Singular Solution

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 Linear Differential Equation of First Order. 

Example: y' − y = y²e^x.

              

Divide both sides of the original differential equation by y²:

      

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

      

      

      

      

A function φ(x) is called the singular solution of the differential equation F(x,y,y') = 0, if uniqueness of solution is violated at each point of the domain of the equation. Geometrically this means that more than one integral curve with the common tangent line passes through each point (x₀,y₀). 

4. Ricatti 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. 

Solution:

y’=a(x)y + b(x) + C(x) (1)

if C=0 => Bernoulli equation

C≠0 => Ricatti equation

z – partial solution of (1)

u = (y-z)

z’ = az+b +c (2)

y=u+z y’=u’ + z’ ===> (1)

u’+z’=a(u+z) + b + C

u’+z’=au+az + b + 2bzu + b + C

u’=au + b + 2bzu

u’=(a+2bz)u + b - Bernoulli equation

Special Case 1: Coefficients a, b, c are constants.

If the coefficients in the Riccatti 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:

This integral can be easily calculated at any values of a, b and c (For more information see "Integration of Rational Functions").

Special 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 y² is a constant, and c(x) is a power function:

This case of Riccati equation has nice solutions!  First of all, if n = 0, we get the Case 1 where the variables are separated and the differential equation can be integrated.  If n = −2, the Riccati equation is converted into a homogeneous equation with help of the substitution y = 1/z and then also can be integrated.  This differential equation can be also solved at

Here the general solution is expressed through cylinder functions.  At all other values of the power n, the solution of the Riccati equation can be expressed through integrals of elementary functions. This fact was discovered by the French mathematician Joseph Liouville (1809-1882) in 1841.