61.2.3 problem 3
Internal
problem
ID
[11930]
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
:
3
Date
solved
:
Sunday, March 30, 2025 at 09:20:07 PM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=y^{2}+a^{2} x^{2}+b x +c \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 369
ode:=diff(y(x),x) = y(x)^2+a^2*x^2+b*x+c;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {-48 \left (x \,a^{2}+\frac {b}{2}\right )^{2} \left (i a^{3}-\frac {1}{3} a^{2} c +\frac {1}{12} b^{2}\right ) c_1 \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +28 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {5}{2}\right ], \frac {i \left (2 x \,a^{2}+b \right )^{2}}{4 a^{3}}\right )+48 \left (i a^{4} x^{2}+i a^{2} b x +\frac {1}{4} i b^{2}-a^{3}\right ) a^{3} c_1 \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +12 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 x \,a^{2}+b \right )^{2}}{4 a^{3}}\right )+24 \left (x \,a^{2}+\frac {b}{2}\right ) \left (\left (-i a^{3}+a^{2} c -\frac {1}{4} b^{2}\right ) \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +20 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 x \,a^{2}+b \right )^{2}}{4 a^{3}}\right )+i a^{3} \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +4 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 x \,a^{2}+b \right )^{2}}{4 a^{3}}\right )\right )}{48 \left (c_1 \left (x \,a^{2}+\frac {b}{2}\right ) \operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +12 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 x \,a^{2}+b \right )^{2}}{4 a^{3}}\right )+\frac {\operatorname {hypergeom}\left (\left [\frac {4 i a^{2} c +4 a^{3}-i b^{2}}{16 a^{3}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 x \,a^{2}+b \right )^{2}}{4 a^{3}}\right )}{2}\right ) a^{4}}
\]
✓ Mathematica. Time used: 0.729 (sec). Leaf size: 664
ode=D[y[x],x]==y[x]^2+a^2*x^2+b*x+c;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(-a**2*x**2 - b*x - c - y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -a**2*x**2 - b*x - c - y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method