Jacopo Riccati

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:

\begin{equation} y' + p(x)\,y = g(x)\,y^2 + h(x) , \label{Eq.riccati.1} \end{equation}
where p, g, and h are some real-valued given functions. Riccati himself was concerned with solutions to so called special Riccati equation
\begin{equation} y' = a\,y^2 + x^{\alpha} . \label{Eq.riccati.2} \end{equation}

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.

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

\begin{equation} y(x) = - \frac{u'}{g(x)\,u(x)} . \label{Eq.riccati.3} \end{equation}
Substitution of the above expression into the Riccati equation yields the second order linear differential equation for u(x):
\begin{equation} u'' + a(x)\, u' (x) + b(x)\, u =0, \qquad \mbox{where} \quad a(x) = p(x) - \frac{g' (x)}{g(x)} , \quad b(x) = g(x)\,h(x) . \label{Eq.riccati.4} \end{equation}
Conversely, every linear homogeneous differential equation with variable coefficients \( u'' + a\,u' + b\,u = 0 \) can be reduced to the Riccati equation
\[ y' + y^2 + a(x)\,y + b(x) =0 \]
by the substitution
\[ u(x) = \exp \left\{ y(x)\,{\text d} x \right\} . \]

It is sometimes possible to find a solution of a Riccati equation by guessing. Without knowing a solution to the Riccati equation, there is no chance of finding its general solution explicitly. If one solution ϕ is known, then substitution w = y - ϕ reduces the Riccati equation to a Bernoulli equation. Another substitution y = ϕ + 1/v also reduces the Riccati equation to a Bernoulli type.

Theorem (Liouville, 1841). The special Riccati equation \( y' = a\,y^2 + b\, x^{\alpha} \) can be integrated in closed form via elementary functions if and only if
\[ \frac{\alpha}{2\alpha +4} \qquad\mbox{is an integer}. \qquad ▣ \]

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

\begin{equation} 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} \label{Eq.riccati.5} \end{equation}
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.
We make transformation y = ur vs, which we substitute into the special Riccati equation:
\[ r\,u^{r-1} u' \,v^s + s\,u^r v^{s-1} v' = a\,u^{2r} v^{2s} + b\, x^2 . \]
Choosing
\[ \begin{split} r\,u^{r-1} u' \,v^s &= b\,x^2 , \\ s\,u^r v^{s-1} v' &= a\,u^{2r} v^{2s} \qquad \Longleftrightarrow \qquad s\,v' = a\, u^r v^{s+1} . \end{split} \]
Differentiation of the latter, we obtain
\[ s\, v'' = ar\, u^{r-1} u' \, v^{s+1} + a \left( s+1 \right) u^r v^s v' . \]
The nonlinearity can now be removed by choosing s = -1. Then using the latter, we get
\[ s\, v'' = ab\, u^{r-1} u' \, v^{s+1} = ab\, x^2 v \qquad\mbox{or} \qquad v'' = -ab\, x^2 v . \]
Since the equation \( v'' + ab\, x^2 v = 0 \) has the general solution expressed through Bessel functions (see section)
\[ v(x) = \sqrt{x} \left[ C_1 J_{1/4} \left( \frac{\sqrt{ab}\,x^2}{2} \right) + C_2 Y_{1/4} \left( \frac{\sqrt{ab}\,x^2}{2} \right) \right] , \]
where C1 and C2 are arbitrary constants, we arrive at the required formula. Using Mathematica
v[x_] = Sqrt[ x]*(k*BesselJ[1/4, alpha*x^2 /2] + BesselY[1/4, alpha*x^2 /2])
y[x_] = Simplify[v'[x]/v[x]]
Assuming[alpha > 0 && x > 0, Series[%, {x, 0, 0}]]
Normal[%]
Simplify[%]
we obtain the general solution of the special Riccati equation:
\[ y(x) = \frac{1}{2x} \,\frac{\sqrt{ab}\,k x^2 J_{3/4} \left( \frac{\sqrt{ab}\, x^2}{2} \right) + k\, J_{1/4} \left( \frac{\sqrt{ab}\, x^2}{2} \right) - \sqrt{ab} \, k x^2 J_{5/4} \left( \frac{\sqrt{ab}\, x^2}{2} \right) + \sqrt{ab}\, x^2 Y_{-3/4} \left( \frac{\sqrt{ab}\, x^2}{2} \right) + Y_{1/4} \left( \frac{\sqrt{ab}\, x^2}{2} \right) - \sqrt{ab} \,x^2 Y_{5/4} \left( \frac{\sqrt{ab}\, x^2}{2} \right) }{k\, J_{1/4} \left( \frac{\sqrt{ab}\, x^2}{2} \right) + Y_{1/4} \left( \frac{\sqrt{ab}\, x^2}{2} \right)} , \]
where k is an arbitrary constant, and ab is positive.    ◂

 

We can find the limit of the solution to the special Riccati equation when x → 0:
\begin{align} \lim_{x\to 0} y(x) &= \frac{\sqrt{ab}}{8\,\Gamma^2 \left( \frac{1}{4} \right) \Gamma \left( \frac{5}{4} \right)} \left[ \sqrt{2}\, \Gamma \left( \frac{1}{4} \right) \left( \Gamma \left( -\frac{1}{4} \right) - 4\,\Gamma \left( \frac{3}{4} \right) \right) \Gamma \left( \frac{5}{4} \right) -2k\pi \left( \Gamma \left( \frac{1}{4} \right) + 4\, \Gamma \left( \frac{5}{4} \right)\right) \right] \notag \\ &= -\frac{\sqrt{ab}}{\pi} \,\Gamma^2 \left( \frac{3}{4} \right) \left( 1+ k\right) \approx -0.477989 \left( 1 + k \right) \sqrt{ab} . \label{Eq.riccati.6} \end{align} 
simplify(sqrt(2)*(GAMMA(-1/4)-4*GAMMA(3/4))/(8*GAMMA(1/4)))
evalf(%)
simplify(pi*(GAMMA(1/4)+4*GAMMA(5/4))/((4*GAMMA(1/4)*GAMMA(1/4))*GAMMA(5/4)))
evalf(%)
\[ \frac{1}{\pi}\,\Gamma \left( \frac{3}{4} \right)^2 \]

Consider the Riccati equation

\[ y' = 2y/x+y^2 -x^4 . \tag{4.1} \]

We plot the separatrix and the direction field

field = StreamPlot[{1, 2*y/x + y^2 - x^4}, {x, -2, 2}, {y, -2, 2}, StreamColorFunction -> "Rainbow", StreamPoints -> 42, StreamScale -> {Full, All, 0.04}];
sol = NDSolve[{y'[x] == 2*y[x]/x + (y[x])^2 - x^4 , y[0.1] == 0.01}, y[x], {x, 0, 2}];
curve = Plot[Evaluate[y[x] /. sol], {x, 0, 2}, PlotRange -> {{0, 2}, {-2, 2}}, PlotStyle -> {Red, Thickness[0.01]}];
Show[field, curve]
         Direction field and separatrix (in red).       Mathematica code.

The given Riccati equation can be solved by substitution \( y =x^2 +1/v(x) , \) where y1 = x² 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])
   ■
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