60.1.138 problem 139
Internal
problem
ID
[10152]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
139
Date
solved
:
Monday, January 27, 2025 at 06:30:11 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} x^{2} \left (y^{\prime }+y^{2}\right )+a \,x^{k}-b \left (b -1\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 229
dsolve(x^2*(diff(y(x),x)+y(x)^2) + a*x^k - b*(b-1)=0,y(x), singsol=all)
\[
y = \frac {-\left (\operatorname {BesselY}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )+k}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )+k}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right ) x^{\frac {k}{2}} \sqrt {a}+\left (\operatorname {BesselY}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right ) \left (\frac {1}{2}+\operatorname {csgn}\left (2 b -1\right ) b -\frac {\operatorname {csgn}\left (2 b -1\right )}{2}\right )}{x \left (\operatorname {BesselY}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right ) c_{1} +\operatorname {BesselJ}\left (\frac {\operatorname {csgn}\left (2 b -1\right ) \left (2 b -1\right )}{k}, \frac {2 \sqrt {a}\, x^{\frac {k}{2}}}{k}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.469 (sec). Leaf size: 627
DSolve[x^2*(D[y[x],x]+y[x]^2) + a*x^k - b*(b-1)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt {a} x^k \operatorname {Gamma}\left (\frac {2 b+k-1}{k}\right ) \operatorname {BesselJ}\left (-\frac {-2 b+k+1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-\sqrt {a} x^k \operatorname {Gamma}\left (\frac {2 b+k-1}{k}\right ) \operatorname {BesselJ}\left (\frac {2 b+k-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )+\sqrt {x^k} \operatorname {Gamma}\left (\frac {2 b+k-1}{k}\right ) \operatorname {BesselJ}\left (\frac {2 b-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-\sqrt {a} c_1 x^k \operatorname {Gamma}\left (\frac {-2 b+k+1}{k}\right ) \operatorname {BesselJ}\left (\frac {-2 b+k+1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )+\sqrt {a} c_1 x^k \operatorname {Gamma}\left (\frac {-2 b+k+1}{k}\right ) \operatorname {BesselJ}\left (-\frac {2 b+k-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )+c_1 \sqrt {x^k} \operatorname {Gamma}\left (\frac {-2 b+k+1}{k}\right ) \operatorname {BesselJ}\left (\frac {1-2 b}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )}{2 x \sqrt {x^k} \left (\operatorname {Gamma}\left (\frac {2 b+k-1}{k}\right ) \operatorname {BesselJ}\left (\frac {2 b-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )+c_1 \operatorname {Gamma}\left (\frac {-2 b+k+1}{k}\right ) \operatorname {BesselJ}\left (\frac {1-2 b}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right )} \\
y(x)\to \frac {\frac {\sqrt {a} \sqrt {x^k} \left (\operatorname {BesselJ}\left (-\frac {2 b+k-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-\operatorname {BesselJ}\left (\frac {-2 b+k+1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right )}{\operatorname {BesselJ}\left (\frac {1-2 b}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )}+1}{2 x} \\
y(x)\to \frac {\frac {\sqrt {a} \sqrt {x^k} \left (\operatorname {BesselJ}\left (-\frac {2 b+k-1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )-\operatorname {BesselJ}\left (\frac {-2 b+k+1}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )\right )}{\operatorname {BesselJ}\left (\frac {1-2 b}{k},\frac {2 \sqrt {a} \sqrt {x^k}}{k}\right )}+1}{2 x} \\
\end{align*}