61.2.37 problem 37
Internal
problem
ID
[12043]
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
:
37
Date
solved
:
Monday, January 27, 2025 at 11:53:15 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x y^{\prime }&=x y^{2}+a y+b \,x^{n} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 139
dsolve(x*diff(y(x),x)=x*y(x)^2+a*y(x)+b*x^n,y(x), singsol=all)
\[
y = \frac {x^{\frac {n}{2}-\frac {1}{2}} \sqrt {b}\, \left (\operatorname {BesselY}\left (\frac {-a +n}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {-a +n}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right )\right )}{\operatorname {BesselY}\left (\frac {-a -1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {-a -1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right )}
\]
✓ Solution by Mathematica
Time used: 0.688 (sec). Leaf size: 855
DSolve[x*D[y[x],x]==x*y[x]^2+a*y[x]+b*x^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}} \operatorname {Gamma}\left (\frac {a+n+2}{n+1}\right ) \operatorname {BesselJ}\left (\frac {a-n}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )-\sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}} \operatorname {Gamma}\left (\frac {a+n+2}{n+1}\right ) \operatorname {BesselJ}\left (\frac {a+n+2}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )+a \operatorname {Gamma}\left (\frac {a+n+2}{n+1}\right ) \operatorname {BesselJ}\left (\frac {a+1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )+\operatorname {Gamma}\left (\frac {a+n+2}{n+1}\right ) \operatorname {BesselJ}\left (\frac {a+1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )-\sqrt {b} c_1 \left (x^n\right )^{\frac {n+1}{2 n}} \operatorname {Gamma}\left (\frac {n-a}{n+1}\right ) \operatorname {BesselJ}\left (\frac {n-a}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )+\sqrt {b} c_1 \left (x^n\right )^{\frac {n+1}{2 n}} \operatorname {Gamma}\left (\frac {n-a}{n+1}\right ) \operatorname {BesselJ}\left (-\frac {a+n+2}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )+(a+1) c_1 \operatorname {Gamma}\left (\frac {n-a}{n+1}\right ) \operatorname {BesselJ}\left (-\frac {a+1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )}{2 x \left (\operatorname {Gamma}\left (\frac {a+n+2}{n+1}\right ) \operatorname {BesselJ}\left (\frac {a+1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )+c_1 \operatorname {Gamma}\left (\frac {n-a}{n+1}\right ) \operatorname {BesselJ}\left (-\frac {a+1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )\right )} \\
y(x)\to -\frac {\frac {\sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}} \left (\operatorname {BesselJ}\left (-\frac {a+n+2}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )-\operatorname {BesselJ}\left (\frac {n-a}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )\right )}{\operatorname {BesselJ}\left (-\frac {a+1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )}+a+1}{2 x} \\
y(x)\to -\frac {\frac {\sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}} \left (\operatorname {BesselJ}\left (-\frac {a+n+2}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )-\operatorname {BesselJ}\left (\frac {n-a}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )\right )}{\operatorname {BesselJ}\left (-\frac {a+1}{n+1},\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )}+a+1}{2 x} \\
\end{align*}