55.2.3 problem 3

Internal problem ID [13229]
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 : Wednesday, October 01, 2025 at 03:39:11 AM
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: 0.633 (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