55.3.4 problem 4
Internal
problem
ID
[13308]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
subsection
1.2.3.
Equations
Containing
Exponential
Functions
Problem
number
:
4
Date
solved
:
Wednesday, October 01, 2025 at 06:41:58 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=\sigma y^{2}+a y+b \,{\mathrm e}^{x}+c \end{align*}
✓ Maple. Time used: 0.016 (sec). Leaf size: 200
ode:=diff(y(x),x) = sigma*y(x)^2+a*y(x)+b*exp(x)+c;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {-2 \,{\mathrm e}^{\frac {x}{2}} \operatorname {BesselJ}\left (\sqrt {a^{2}-4 \sigma c}+1, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) \sqrt {b}\, \sigma -2 \,{\mathrm e}^{\frac {x}{2}} \operatorname {BesselY}\left (\sqrt {a^{2}-4 \sigma c}+1, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) \sqrt {b}\, c_1 \sigma +\sqrt {\sigma }\, \left (\operatorname {BesselY}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_1 +\operatorname {BesselJ}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right )\right ) \left (\sqrt {a^{2}-4 \sigma c}+a \right )}{\sigma ^{{3}/{2}} \left (2 \operatorname {BesselY}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_1 +2 \operatorname {BesselJ}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right )\right )}
\]
✓ Mathematica. Time used: 0.337 (sec). Leaf size: 546
ode=D[y[x],x]==sigma*y[x]^2+a*y[x]+b*Exp[x]+c;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {a \sqrt {b \sigma e^x} \operatorname {Gamma}\left (\sqrt {a^2-4 c \sigma }+1\right ) \operatorname {BesselJ}\left (\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )+b \sigma e^x \operatorname {Gamma}\left (\sqrt {a^2-4 c \sigma }+1\right ) \operatorname {BesselJ}\left (\sqrt {a^2-4 c \sigma }-1,2 \sqrt {b e^x \sigma }\right )-b \sigma e^x \operatorname {Gamma}\left (\sqrt {a^2-4 c \sigma }+1\right ) \operatorname {BesselJ}\left (\sqrt {a^2-4 c \sigma }+1,2 \sqrt {b e^x \sigma }\right )+a c_1 \sqrt {b \sigma e^x} \operatorname {Gamma}\left (1-\sqrt {a^2-4 c \sigma }\right ) \operatorname {BesselJ}\left (-\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )+b c_1 \sigma e^x \operatorname {Gamma}\left (1-\sqrt {a^2-4 c \sigma }\right ) \operatorname {BesselJ}\left (-\sqrt {a^2-4 c \sigma }-1,2 \sqrt {b e^x \sigma }\right )-b c_1 \sigma e^x \operatorname {Gamma}\left (1-\sqrt {a^2-4 c \sigma }\right ) \operatorname {BesselJ}\left (1-\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )}{2 \sigma \sqrt {b \sigma e^x} \left (\operatorname {Gamma}\left (\sqrt {a^2-4 c \sigma }+1\right ) \operatorname {BesselJ}\left (\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )+c_1 \operatorname {Gamma}\left (1-\sqrt {a^2-4 c \sigma }\right ) \operatorname {BesselJ}\left (-\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )\right )}\\ y(x)&\to \frac {\frac {\sqrt {b \sigma e^x} \left (\operatorname {BesselJ}\left (1-\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )-\operatorname {BesselJ}\left (-\sqrt {a^2-4 c \sigma }-1,2 \sqrt {b e^x \sigma }\right )\right )}{\operatorname {BesselJ}\left (-\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )}-a}{2 \sigma } \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
sigma = symbols("sigma")
y = Function("y")
ode = Eq(-a*y(x) - b*exp(x) - c - sigma*y(x)**2 + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -a*y(x) - b*exp(x) - c - sigma*y(x)**2 + Derivative(y(x), x) cannot be solved by the lie group method