Differential Equations

1st Order Equations

Riccati Equation

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Problems 1-4

General Riccati Equation

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

\[{y’ = a\left( x \right)y + b\left( x \right){y^2} }+{ c\left( x \right),}\]

where \(a\left( x \right),\) \(b\left( x \right),\) \(c\left( x \right)\) are continuous functions of \(x.\)

The Riccati equation is used in different areas of mathematics (for example, in algebraic geometry and the theory of conformal mapping), and physics. It also appears in many applied problems.

The differential equation given above is called the general Riccati equation. It 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

\[y = {y_1} + u.\]

Indeed, substituting the solution \(y = {y_1} + u\) into Riccati equation, we have

{{\left( {{y_1} + u} \right)^\prime } }
= {a\left( x \right)\left( {{y_1} + u} \right) }+{ b\left( x \right){\left( {{y_1} + u} \right)^2} }+{ c\left( x \right),}
{\underline {{y_1}^\prime } + u’ }
= {\underline {a\left( x \right){y_1}} + a\left( x \right)u }+{ \underline {b\left( x \right)y_1^2} + 2b\left( x \right){y_1}u }+{ b\left( x \right){u^2} }+{ \underline {c\left( x \right)} .}

The underlined terms in the left and in the right side can be canceled because \({y_1}\) is a particular solution satisfying the equation. As a result we obtain the differential equation for the function \(u\left( x \right):\)

\[{u’ = b\left( x \right){u^2} }+{ \left[ {2b\left( x \right){y_1} + a\left( x \right)} \right]u,}\]

which is a Bernoulli equation.

Substitution of \(z = {\large\frac{1}{u}\normalsize}\) converts the given Bernoulli equation into a linear differential equation that
allows integration.

Besides the general Riccati equation, there is an infinite number of particular cases of Riccati equation at certain coefficients of \(a\left( x \right),\) \(b\left( x \right),\) and \(c\left( x \right).\) Many of these particular cases have integrable solutions.

Returning to the general Riccati equation, we see that we can construct the general solution if a particular solution is known. Unfortunately, there is no sctrict algorithm to find the particular solution, which depends on the types of the functions \(a\left( x \right),\) \(b\left( x \right),\) and \(c\left( x \right).\)

Below we consider some well known particular cases of the Riccati equation.

Special 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:

{y’ = ay + b{y^2} + c,\;\;}\Rightarrow
{\frac{{dy}}{{dx}} = ay + b{y^2} + c,\;\;}\Rightarrow
{\int {\frac{{dy}}{{ay + b{y^2} + c}}} = \int {dx} .}

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 \(y’ = b{y^2} + c{x^n}\)

Consider a Riccati equation of type \(y’ = b{y^2} + c{x^n},\) where the function \(a\left( x \right)\) at the linear term is zero, the coefficient \(b\) at \({y^2}\) is a constant, and \(c\left( x \right)\) is a power function:

\[{a\left( x \right) \equiv 0,\;\;}\kern-0.3pt{b\left( x \right) = b,\;\;}\kern0pt{c\left( x \right) = c{x^n}.}\]

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 \(y = {\large\frac{1}{z}\normalsize}\) and then also can be integrated.

This differential equation can be also solved at

\[{n = \frac{{4k}}{{1 – 2k}},\;\;}\kern-0.3pt{\text{where}\;\;}\kern0pt{k = \pm 1, \pm 2, \pm 3, \ldots }\]

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 \(\left( {1809 – 1882} \right)\) in \(1841.\)

Solved Problems

Click on problem description to see solution.

 Example 1

Solve the differential equation \(y’ = y + {y^2} + 1.\)

 Example 2

Solve the Riccati equation \(y’ + {y^2} =\) \( {\large\frac{2}{{{x^2}}}\normalsize}.\)

 Example 3

Find the general solution of the differential equation \({x^3}y’ + {x^2}y \) \(- {y^2} \) \(= 2{x^4}.\)

 Example 4

Solve the equation \(y’ + 6{y^2} \) \(= {\large\frac{1}{{{x^2}}}\normalsize}.\)

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Problems 1-4