55.2.20 problem 20
Internal
problem
ID
[13246]
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
:
20
Date
solved
:
Wednesday, October 01, 2025 at 04:07:19 AM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} \left (a \,x^{2}+b x +c \right )^{2} \left (y^{\prime }+y^{2}\right )+A&=0 \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 481
ode:=(a*x^2+b*x+c)^2*(diff(y(x),x)+y(x)^2)+A = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {2 a \left (\left (i \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}\, a \sqrt {4 a c -b^{2}}-2 \left (x a +\frac {b}{2}\right ) \sqrt {-4 a c +b^{2}}\right ) c_1 {\left (\frac {i \sqrt {4 a c -b^{2}}-2 x a -b}{2 x a +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}-{\left (\frac {i \sqrt {4 a c -b^{2}}-2 x a -b}{2 x a +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} \left (i \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}\, a \sqrt {4 a c -b^{2}}+2 \left (x a +\frac {b}{2}\right ) \sqrt {-4 a c +b^{2}}\right )\right )}{\sqrt {-4 a c +b^{2}}\, \left (2 x a +b +i \sqrt {4 a c -b^{2}}\right ) \left (i \sqrt {4 a c -b^{2}}-2 x a -b \right ) \left (c_1 {\left (\frac {i \sqrt {4 a c -b^{2}}-2 x a -b}{2 x a +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+{\left (\frac {i \sqrt {4 a c -b^{2}}-2 x a -b}{2 x a +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right )}
\]
✓ Mathematica. Time used: 1.49 (sec). Leaf size: 397
ode=(a*x^2+b*x+c)^2*(D[y[x],x]+y[x]^2)+A==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {2 (x (a x+b)+c) \exp \left (-2 \int _1^x\frac {b+2 a K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (c+K[1] (b+a K[1]))}dK[1]\right )+\left (-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}+2 a x+b\right ) \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {b+2 a K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (c+K[1] (b+a K[1]))}dK[1]\right )dK[2]+c_1 \left (-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}+2 a x+b\right )}{2 (x (a x+b)+c) \left (\int _1^x\exp \left (-2 \int _1^{K[2]}\frac {b+2 a K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (c+K[1] (b+a K[1]))}dK[1]\right )dK[2]+c_1\right )}\\ y(x)&\to \frac {-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}+2 a x+b}{2 (x (a x+b)+c)} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(A + (y(x)**2 + Derivative(y(x), x))*(a*x**2 + b*x + c)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (-A - a**2*x**4*y(x)**2 - 2*a*b*x**3*y(x)**2 - 2*a*c*x**2*y(x)**2 - b**2*x**2*y(x)**2 - 2*b*c*x*y(x)**2 - c**2*y(x)**2)/(a**2*x**4 + 2*a*b*x**3 + 2*a*c*x**2 + b**2*x**2 + 2*b*c*x + c**2) cannot be solved by the factorable group method