Preface

This is a tutorial made solely for the purpose of education and it was designed for students taking Applied Math 0330. It is primarily for students who have very little experience or have never used Mathematica before and would like to learn more of the basics for this computer algebra system. As a friendly reminder, don't forget to clear variables in use and/or the kernel.

Finally, the commands in this tutorial are all written in bold black font, while Mathematica output is in regular fonts. This means that you can copy and paste all comamnds into Mathematica, change the parameters and run them. You, as the user, are free to use the scripts to your needs for learning how to use the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately.

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Riccati Equations

Jacopo Francesco Riccati (1676-1754) was an Venetian mathematician and jurist from Venice. He is best known for having studied the differential equation which bears his name:

\[ y' + p(x)\,y = g(x)\,y^2 + h(x) , \]
where p, g, and h are some real-valued given functions. Riccati himself was concerned with solutions to so called special Riccati equation
\[ y' = a\,y^2 + x^{\alpha} . \]

Riccati was educated first at the Jesuit school for the nobility in Brescia, and in 1693 he entered the University of Padua to study law. He received a doctorate in law in 1696. Encouraged by Stefano degli Angeli to pursue mathematics, he studied mathematical analysis. Riccati received various academic offers, but declined them in order to devote his full attention to the study of mathematical analysis on his own. Peter the Great invited him to Russia as president of the St. Petersburg Academy of Sciences. He was also invited to Vienna as an imperial councilor and was offered a professorship at the University of Padua. He declined all these offers. He was often consulted by the Senate of Venice on the construction of canals and dikes along rivers. Methods of Solution of the Riccati Differential Equation D. Robert Haaheim and F. Max Stein Mathematics Magazine Vol. 42, No. 5 (Nov., 1969), pp. 233-240

When h(x) = 0, we get a Bernoulli equation. The Riccati equation has much in common with linear equations; for example, it has no singular solution. Except special cases, the Riccati equation cannot be solved analytically using elementary functions or quadratures, and the most common way to obtain its solution is to represent it in series. Moreover, the Riccati equation can be reduced to the second order linear differential equation by substitution

\[ y(x) = - \frac{u'}{g(x)\,u(x)} . \]

Theorem (Liouville, 1841). The special Ricacti equation \( y' = a\,y^2 + b\, x^{\alpha} \) can be integrated in closed form if and only if

\[ \frac{\alpha}{2\alpha +4} \qquad\mbox{is an integer}. \qquad ■ \]

The special Ricacti equation can be represented as \( y' = -u' /(au) , \) where

\[ u(x) = \sqrt{x}\,\begin{cases} C_1 J_{1/2q} \left( \frac{\sqrt{ab}}{q} \, x^q \right) + C_2 Y_{1/2q} \left( \frac{\sqrt{ab}}{q} \, x^q \right) , & \quad \mbox{if } ab> 0, \\ C_1 I_{1/2q} \left( \frac{\sqrt{-ab}}{q} \, x^q \right) + C_2 K_{1/2q} \left( \frac{\sqrt{-ab}}{q} \, x^q \right) , & \quad \mbox{if } ab< 0, \end{cases} \]
where \( q= 1+ \alpha /2 \) and J(t), Y(t) are Bessel functions, while I(t), K(t) are modified Bessel functions. Note that the general solution depends on the ratio \( C_1 / C_2 \) of two arbitrary constants.

Example: The Riccati equation

\[ y' = x^2 + y^2 \]
has the general solution \( y= -u' /u , \) where
\[ u(x) = \sqrt{x} \left[ C_1 J_{1/4} \left( x^2 /2 \right) + C_2 Y_{1/4} \left( x^2 /2 \right) \right] , \]
where C1 and C2 are arbitrary constants. Similarly, the Riccati equation
\[ y' = x^2 - y^2 \]
is expressed as \( y= u' /u , \) where
\[ u(x) = \sqrt{x} \left[ C_1 I_{1/4} \left( x^2 /2 \right) + C_2 K_{1/4} \left( x^2 /2 \right) \right] , \]

Example: Consider the Riccati equation

\[ y' = 2y/x+y^2 -x^4 . \]

It can be solved by substitution y =x^2 +1/v(x) , where y1=x^2 is a particular solution of the given Riccati equation.

R[x_, y_] = (y'[x] - 2 y[x]/x - y[x]^2 + x^4 )
y1[x_] = x^2
R[x, y1]
Simplify[Expand[v[x]^2 R[x, Function[t, t + t/v[t]]]]]
DSolve[% == 0, v[x], x] (* solve linear equation for v *)
y2[x_] = Simplify[(y1[x] + 1/v[x]) /. %[[1]]]
Out[11]= x^4 - (2 y[x])/x - y[x]^2 + Derivative[1][y][x]
Out[12]= x^2
Out[13]= 0
Out[14]= -(1 + 2 x^2) v[x] + (-1 - x^2 + x^4) v[x]^2 - x (x + Derivative[1][v][x])
Out[15]= {{v[x] -> -(x (-(E^((2 x^3)/3 - 1/6 x^2 (3 + 2 x))/(2 x)) + (
E^((2 x^3)/3 - 1/6 x^2 (3 + 2 x)) (1 + x))/(2 x^2) - (
E^((2 x^3)/3 - 1/6 x^2 (3 + 2 x)) (1 + x) ((5 x^2)/3 - 1/3 x (3 + 2 x)))/(
2 x) + (-((E^(-(1/6) x^2 (3 + 2 x)) (-1 + x))/x^2) +
E^(-(1/6) x^2 (3 + 2 x))/x + ( E^(-(1/6) x^2 (3 + 2 x)) (-1 + x) (-(x^2/3) -
1/3 x (3 + 2 x)))/x) C[1]))/((-1 - x^2 + x^4) (-((E^((2 x^3)/3 - 1/6 x^2 (3 + 2 x)) (1 + x))/(
2 x)) + (E^(-(1/6) x^2 (3 + 2 x)) (-1 + x) C[1])/x))}}
Out[16]= (E^((2 x^3)/3) (-1 - x + x^2) + 2 (-1 + x + x^2) C[1])/(E^((
2 x^3)/3) + 2 C[1])

 

 

First Order ODEs

Plotting Solutions to ODEs

Direction Fields

Separable Equations

Equations Reducible to the Separable Equations

Equations with Linear Fractions

Exact Equations

Integrating Factors

Linear Equations

RC circuits

Bernoulli Equations

Riccati Equations

Existence and Uniqueness

Qualitative Analysis

Bifurcations

Orthogonal Trajectories

Population Models

Applications

 

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