61.3.4 problem 4
Internal
problem
ID
[12088]
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
:
Tuesday, January 28, 2025 at 12:21:37 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=\sigma y^{2}+a y+b \,{\mathrm e}^{x}+c \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 200
dsolve(diff(y(x),x)=sigma*y(x)^2+a*y(x)+b*exp(x)+c,y(x), singsol=all)
\[
y = -\frac {-2 \operatorname {BesselJ}\left (\sqrt {a^{2}-4 \sigma c}+1, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) {\mathrm e}^{\frac {x}{2}} \sqrt {b}\, \sigma -2 \operatorname {BesselY}\left (\sqrt {a^{2}-4 \sigma c}+1, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) {\mathrm e}^{\frac {x}{2}} \sqrt {b}\, c_{1} \sigma +\sqrt {\sigma }\, \left (\operatorname {BesselY}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_{1} +\operatorname {BesselJ}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right )\right ) \left (\sqrt {a^{2}-4 \sigma c}+a \right )}{\sigma ^{{3}/{2}} \left (2 \operatorname {BesselY}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_{1} +2 \operatorname {BesselJ}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.536 (sec). Leaf size: 546
DSolve[D[y[x],x]==sigma*y[x]^2+a*y[x]+b*Exp[x]+c,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*}