61.2.56 problem 56
Internal
problem
ID
[11983]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
1.2.2.
Equations
Containing
Power
Functions
Problem
number
:
56
Date
solved
:
Wednesday, March 05, 2025 at 03:21:33 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} \left (a \,x^{2}+b \right ) y^{\prime }+\alpha y^{2}+\beta x y+\frac {b \left (a +\beta \right )}{\alpha }&=0 \end{align*}
✓ Maple. Time used: 0.023 (sec). Leaf size: 517
ode:=(a*x^2+b)*diff(y(x),x)+alpha*y(x)^2+beta*x*y(x)+b/alpha*(a+beta) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {a^{2} b \left (-\frac {\left (-\frac {-a x +\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \left (a \,x^{2}+b \right ) \left (a \,x^{2}+2 \sqrt {-a b}\, x -b \right ) \operatorname {HeunCPrime}\left (0, -1-\frac {\beta }{a}, 1+\frac {\beta }{2 a}, 0, \frac {1}{2}+\frac {\beta }{2 a}+\frac {\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-a b}}{-a x +\sqrt {-a b}}\right )}{2}-2 c_{1} a \left (\left (3 a \,x^{2}-b \right ) \sqrt {-a b}+a x \left (a \,x^{2}-3 b \right )\right ) b \operatorname {HeunCPrime}\left (0, \frac {\beta }{a}+1, 1+\frac {\beta }{2 a}, 0, \frac {1}{2}+\frac {\beta }{2 a}+\frac {\beta ^{2}}{4 a^{2}}, \frac {2 \sqrt {-a b}}{-a x +\sqrt {-a b}}\right )+\left (a \,x^{2}+b \right ) \left (\frac {\left (-\frac {-a x +\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \left (a \,x^{2}-2 \sqrt {-a b}\, x -b \right ) \operatorname {hypergeom}\left (\left [1, -\frac {\beta }{2 a}\right ], \left [-\frac {\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right )}{4}+\left (\left (-a^{2} x^{2}+\left (-x^{2} \beta -2 b \right ) a -b \beta \right ) \sqrt {-a b}+a^{2} b x \right ) c_{1} \left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{\frac {\beta }{2 a}}\right )\right )}{\left (-\frac {\sqrt {-a b}\, \left (-\frac {-a x +\sqrt {-a b}}{2 \sqrt {-a b}}\right )^{\frac {\beta }{a}} \left (a \,x^{2}+b \right ) \operatorname {hypergeom}\left (\left [1, -\frac {\beta }{2 a}\right ], \left [-\frac {\beta }{a}\right ], \frac {2 \sqrt {-a b}}{a x +\sqrt {-a b}}\right )}{4}+\left (\frac {a x -\sqrt {-a b}}{a x +\sqrt {-a b}}\right )^{\frac {\beta }{2 a}} a^{2} b c_{1} \left (-\sqrt {-a b}\, x +b \right )\right ) \left (a x -\sqrt {-a b}\right )^{2} \alpha }
\]
✓ Mathematica. Time used: 0.662 (sec). Leaf size: 27
ode=(a*x^2+b)*D[y[x],x]+\[Alpha]*y[x]^2+\[Beta]*x*y[x]+b/\[Alpha]*(a+\[Beta])==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {x (a+\beta )}{\alpha } \\
y(x)\to -\frac {x (a+\beta )}{\alpha } \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
Alpha = symbols("Alpha")
BETA = symbols("BETA")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(Alpha*y(x)**2 + BETA*x*y(x) + (a*x**2 + b)*Derivative(y(x), x) + b*(BETA + a)/Alpha,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out