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:
\[
y' + p(x)\,y = g(x)\,y^2 + h(x) ,
\]
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.
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
\[
y(x) = - \frac{u'}{g(x)\,u(x)} .
\]
Substitution of the above expression into the Riccati equation yields the second order linear differential equation for
u(
x):
\[
u'' + a(x)\, u' (x) + b(x)\, u =0, \qquad \mbox{where} \quad a(x) = p(x) - \frac{g' (x)}{g(x)} , \quad b(x) = g(x)\,h(x) .
\]
Conversely, every linear homogeneous differential equation with variable coefficients
\( u'' + a\,u' + b\,u = 0 \) can be reduced to the Riccari 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 Ricati 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 Ricacti 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 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.
We make transformation
y =
ur vs, which we substitute into the special Ricacti 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 Ricacti 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 Ricacti 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]
\\
&= -\frac{\sqrt{ab}}{\pi} \,\Gamma^2 \left( \frac{3}{4} \right) \left( 1+ k\right) \approx -0.477989 \left( 1 + k \right) \sqrt{ab} .
\end{align*}
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.
The solution of the above Riccati equation subject to the homogeneous initial condition
y(0) = 0 is
\[
y(x) = x\,\frac{-Y_{-3/4} \left( \frac{x^2}{2} \right) + J_{-3/4} \left( \frac{x^2}{2} \right)}{Y_{1/4} \left( \frac{x^2}{2} \right) - J_{1/4} \left( \frac{x^2}{2} \right)} .
\]
This solution blows up at
x ≈ 2.00315 where the denominator is zero.
FindRoot[BesselY[1/4, x^2/2] - BesselJ[1/4, x^2/2] == 0, {x, 2}]
2.003147359426885
When you try to find the solution of the above initial value problem with
Mathematica by entering
DSolve[{y'[x] == x^2 + (y[x])^2, y[0] == 0}, y[x], x]
DSolve::bvnul: For some branches of the general solution, the given boundary conditions lead to an empty solution.
input picture
Looking at the output, we find another solution of the given initialy value problem \( y' = y^2 + x^2 , \quad y(0) =0 \) to be
expressed through Bessel functions of the first find only:
\[
y(x) = x\,\frac{J_{3/4} \left( \frac{x^2}{2} \right)}{J_{-1/4} \left( \frac{x^2}{2} \right)} .
\]
The solution of the above Riccati equation \( y' = y^2 + x^2 \) subject to the initial condition y(0) = 1 is
\[
y(x) = x\,\frac{\left( \pi - \Gamma^2 \left( \frac{3}{4} \right) \right) J_{-3/4} \left( \frac{x^2}{2} \right) - \Gamma^2 \left( \frac{3}{4} \right) Y_{-3/4} \left( \frac{x^2}{2} \right)}{\left( \pi - \Gamma^2 \left( \frac{3}{4} \right) \right) J_{1/4} \left( \frac{x^2}{2} \right) + \Gamma^2 \left( \frac{3}{4} \right) Y_{1/4} \left( \frac{x^2}{2} \right)} .
\]
This solution blows up at
x ≈ 0.969811 where the denominator is zero.
DSolve[{u'[x] == x^2 + (u[x])^2 , u[0] == 1}, u, x]
FindRoot[BesselJ[1/4, x^2/2] Gamma[1/4] -
2 BesselJ[-(1/4), x^2/2] Gamma[3/4] == 0, {x, 1}]
{x -> 0.969811}
The solution of the initial value problem
\[
y' = x^2 + y^2 , \qquad y(0) = -1 ,
\]
is
\[
y(x) = x\,\frac{\left( \pi + \Gamma^2 \left( \frac{3}{4} \right) \right) J_{-3/4} \left( \frac{x^2}{2} \right) - \Gamma^2 \left( \frac{3}{4} \right) Y_{-3/4} \left( \frac{x^2}{2} \right)}{\left( \pi + \Gamma^2 \left( \frac{3}{4} \right) \right) J_{1/4} \left( \frac{x^2}{2} \right) + \Gamma^2 \left( \frac{3}{4} \right) Y_{1/4} \left( \frac{x^2}{2} \right)} .
\]
This solution blows up at
x ≈ 4.1785 where the denominator is zero.
DSolve[{u'[x] == x^2 + (u[x])^2 , u[0] == -1}, u, x]
FindRoot[
BesselJ[1/4, x^2/2] Gamma[1/4] +
2 BesselJ[-(1/4), x^2/2] Gamma[3/4] == 0, {x, 4}]
{x -> 4.17851}
Similarly, the general solution to the Riccati equation
\[
y' = x^2 - y^2
\]
is expressed as
\( y= u' /u , \) where
\[
u(x) = \sqrt{x} \left[ C_1 I_{1/4} \left( x^2 /2 \right) + C_2 K_{1/4} \left( x^2 /2 \right) \right] ,
\]
When we impose the homogeneous conditions
y(0) = 0, then its solution becomes
\[
y(x) = x\,\frac{I_{-3/4} \left( \frac{x^2}{2} \right) \pi \sqrt{2} - 2\,K_{3/4} \left( \frac{x^2}{2} \right)}{\pi \sqrt{2}\,I_{1/4} \left( \frac{x^2}{2} \right) + 2\, K_{1/4} \left( \frac{x^2}{2} \right)} .
\]
The initial value problem
\[
y' = x^2 - y^2 , \qquad y(0) =1 ,
\]
has the solution
\[
y(x) = x\,\frac{\left( \pi + \Gamma^2 \left( \frac{3}{4} \right) \sqrt{2} \right) I_{-3/4} \left( \frac{x^2}{2} \right) \pi - 2\, \Gamma^2 \left( \frac{3}{4} \right) K_{3/4} \left( \frac{x^2}{2} \right)}{\left( \pi + \Gamma^2 \left( \frac{3}{4} \right) \sqrt{2} \right) I_{1/4} \left( \frac{x^2}{2} \right) \pi + 2\,\Gamma^2 \left( \frac{3}{4} \right) K_{1/4} \left( \frac{x^2}{2} \right)} .
\]
■
Click to view the code!
figure(1)
plot(x, y, 'Color', [1 0 0]) %blue line
hold on
plot(x, z, 'Color', [0 1 0]) %green line
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 = 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])
■
figure(1)
plot(x, y, 'Color', [1 0 0]) %blue line
hold on
plot(x, z, 'Color', [0 1 0]) %green line
-
Haaheim, D.R. and Stein, F.M.,
Methods of Solution of the Riccati Differential Equation,
Mathematics Magazine, 1969,
Vol. 42, No. 5, pp. 233--240; https://doi.org/10.1080/0025570X.1969.11975969
-
Robin, W., A new Riccati equation arising in the theory of classical orthogonal polynomials, International Journal of Mathematical Education in Science and Technology, 2003,
Volume 34, 2003 - Issue 1, pp. 31--42. https://doi.org/10.1080/0020739021000018746
-