61.3.1 problem 1
Internal
problem
ID
[12085]
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
:
1
Date
solved
:
Tuesday, January 28, 2025 at 12:21:30 AM
CAS
classification
:
[_Riccati]
\begin{align*} y^{\prime }&=a y^{2}+b \,{\mathrm e}^{\lambda x} \end{align*}
✓ Solution by Maple
Time used: 0.001 (sec). Leaf size: 96
dsolve(diff(y(x),x)=a*y(x)^2+b*exp(lambda*x),y(x), singsol=all)
\[
y = \frac {\sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}} \left (\operatorname {BesselY}\left (1, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right ) c_{1} +\operatorname {BesselJ}\left (1, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )\right )}{\sqrt {a}\, \left (c_{1} \operatorname {BesselY}\left (0, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )+\operatorname {BesselJ}\left (0, \frac {2 \sqrt {a}\, \sqrt {b}\, {\mathrm e}^{\frac {\lambda x}{2}}}{\lambda }\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.339 (sec). Leaf size: 197
DSolve[D[y[x],x]==a*y[x]^2+b*Exp[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt {b e^{\lambda x}} \left (2 \operatorname {BesselY}\left (1,\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )+c_1 \operatorname {BesselJ}\left (1,\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )\right )}{\sqrt {a} \left (2 \operatorname {BesselY}\left (0,\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )+c_1 \operatorname {BesselJ}\left (0,\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )\right )} \\
y(x)\to \frac {\sqrt {b e^{\lambda x}} \operatorname {BesselJ}\left (1,\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )}{\sqrt {a} \operatorname {BesselJ}\left (0,\frac {2 \sqrt {a} \sqrt {b e^{x \lambda }}}{\lambda }\right )} \\
\end{align*}