61.2.4 problem 4
Internal
problem
ID
[12010]
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
:
4
Date
solved
:
Monday, January 27, 2025 at 11:48:24 PM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} y^{\prime }&=a y^{2}+b \,x^{n} \end{align*}
✓ Solution by Maple
Time used: 0.017 (sec). Leaf size: 207
dsolve(diff(y(x),x)=a*y(x)^2+b*x^n,y(x), singsol=all)
\[
y = \frac {\sqrt {a b}\, x^{\frac {n}{2}+1} \operatorname {BesselJ}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right ) c_{1} +\operatorname {BesselY}\left (\frac {n +3}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right ) \sqrt {a b}\, x^{\frac {n}{2}+1}-c_{1} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )-\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )}{x a \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )+\operatorname {BesselY}\left (\frac {1}{n +2}, \frac {2 \sqrt {a b}\, x^{\frac {n}{2}+1}}{n +2}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.416 (sec). Leaf size: 752
DSolve[D[y[x],x]==a*y[x]^2+b*x^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt {a} \sqrt {b} x^{\frac {n}{2}+1} \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2}-1,\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )-\sqrt {a} \sqrt {b} x^{\frac {n}{2}+1} \operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (1+\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+\operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )-\sqrt {a} \sqrt {b} c_1 x^{\frac {n}{2}+1} \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+\sqrt {a} \sqrt {b} c_1 x^{\frac {n}{2}+1} \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )}{2 a x \left (\operatorname {Gamma}\left (1+\frac {1}{n+2}\right ) \operatorname {BesselJ}\left (\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )+c_1 \operatorname {Gamma}\left (\frac {n+1}{n+2}\right ) \operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )\right )} \\
y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} x^{n/2} \left (\operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )-\operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )\right )}{\operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )}-\frac {1}{x}}{2 a} \\
y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} x^{n/2} \left (\operatorname {BesselJ}\left (\frac {n+1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )-\operatorname {BesselJ}\left (-\frac {n+3}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )\right )}{\operatorname {BesselJ}\left (-\frac {1}{n+2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {n}{2}+1}}{n+2}\right )}-\frac {1}{x}}{2 a} \\
\end{align*}