61.2.35 problem 35
Internal
problem
ID
[12041]
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
:
35
Date
solved
:
Monday, January 27, 2025 at 11:53:09 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x y^{\prime }&=a y^{2}+b y+c \,x^{n} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 164
dsolve(x*diff(y(x),x)=a*y(x)^2+b*y(x)+c*x^n,y(x), singsol=all)
\[
y = \frac {\left (\operatorname {BesselY}\left (\frac {b +n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b +n}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right ) \sqrt {a c}\, x^{\frac {n}{2}}-b \left (\operatorname {BesselY}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )}{a \left (\operatorname {BesselY}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {b}{n}, \frac {2 \sqrt {a c}\, x^{\frac {n}{2}}}{n}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.304 (sec). Leaf size: 402
DSolve[x*D[y[x],x]==a*y[x]^2+b*y[x]+c*x^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt {a} \sqrt {c} x^{n/2} \left (-2 \operatorname {BesselJ}\left (\frac {b}{n}-1,\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )+c_1 \left (\operatorname {BesselJ}\left (1-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )-\operatorname {BesselJ}\left (-\frac {b+n}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )\right )\right )-b c_1 \operatorname {BesselJ}\left (-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )}{2 a \left (\operatorname {BesselJ}\left (\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )+c_1 \operatorname {BesselJ}\left (-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )\right )} \\
y(x)\to -\frac {-\sqrt {a} \sqrt {c} x^{n/2} \operatorname {BesselJ}\left (1-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )+\sqrt {a} \sqrt {c} x^{n/2} \operatorname {BesselJ}\left (-\frac {b+n}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )+b \operatorname {BesselJ}\left (-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )}{2 a \operatorname {BesselJ}\left (-\frac {b}{n},\frac {2 \sqrt {a} \sqrt {c} x^{n/2}}{n}\right )} \\
\end{align*}