60.1.24 problem 24
Internal
problem
ID
[10038]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
24
Date
solved
:
Monday, January 27, 2025 at 06:19:08 PM
CAS
classification
:
[[_Riccati, _special]]
\begin{align*} y^{\prime }+a y^{2}-b \,x^{\nu }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 214
dsolve(diff(y(x),x) + a*y(x)^2 - b*x^nu=0,y(x), singsol=all)
\[
y = \frac {-\sqrt {-a b}\, x^{\frac {\nu }{2}+1} \operatorname {BesselJ}\left (\frac {3+\nu }{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right ) c_{1} -\operatorname {BesselY}\left (\frac {3+\nu }{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right ) \sqrt {-a b}\, x^{\frac {\nu }{2}+1}+c_{1} \operatorname {BesselJ}\left (\frac {1}{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\operatorname {BesselY}\left (\frac {1}{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )}{x a \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\operatorname {BesselY}\left (\frac {1}{\nu +2}, \frac {2 \sqrt {-a b}\, x^{\frac {\nu }{2}+1}}{\nu +2}\right )\right )}
\]
✓ Solution by Mathematica
Time used: 0.408 (sec). Leaf size: 770
DSolve[D[y[x],x] + a*y[x]^2 - b*x^nu == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt {a} \sqrt {b} (-1)^{\frac {1}{\nu +2}} x^{\frac {\nu }{2}+1} \operatorname {Gamma}\left (1+\frac {1}{\nu +2}\right ) \operatorname {BesselI}\left (\frac {1}{\nu +2}-1,\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\sqrt {a} \sqrt {b} (-1)^{\frac {1}{\nu +2}} x^{\frac {\nu }{2}+1} \operatorname {Gamma}\left (1+\frac {1}{\nu +2}\right ) \operatorname {BesselI}\left (1+\frac {1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+(-1)^{\frac {1}{\nu +2}} \operatorname {Gamma}\left (1+\frac {1}{\nu +2}\right ) \operatorname {BesselI}\left (\frac {1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\sqrt {a} \sqrt {b} c_1 x^{\frac {\nu }{2}+1} \operatorname {Gamma}\left (\frac {\nu +1}{\nu +2}\right ) \operatorname {BesselI}\left (\frac {\nu +1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\sqrt {a} \sqrt {b} c_1 x^{\frac {\nu }{2}+1} \operatorname {Gamma}\left (\frac {\nu +1}{\nu +2}\right ) \operatorname {BesselI}\left (-\frac {\nu +3}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+c_1 \operatorname {Gamma}\left (\frac {\nu +1}{\nu +2}\right ) \operatorname {BesselI}\left (-\frac {1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )}{2 a x \left ((-1)^{\frac {1}{\nu +2}} \operatorname {Gamma}\left (1+\frac {1}{\nu +2}\right ) \operatorname {BesselI}\left (\frac {1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+c_1 \operatorname {Gamma}\left (\frac {\nu +1}{\nu +2}\right ) \operatorname {BesselI}\left (-\frac {1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )\right )} \\
y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} x^{\nu /2} \left (\operatorname {BesselI}\left (\frac {\nu +1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\operatorname {BesselI}\left (-\frac {\nu +3}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )\right )}{\operatorname {BesselI}\left (-\frac {1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )}+\frac {1}{x}}{2 a} \\
y(x)\to \frac {\frac {\sqrt {a} \sqrt {b} x^{\nu /2} \left (\operatorname {BesselI}\left (\frac {\nu +1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )+\operatorname {BesselI}\left (-\frac {\nu +3}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )\right )}{\operatorname {BesselI}\left (-\frac {1}{\nu +2},\frac {2 \sqrt {a} \sqrt {b} x^{\frac {\nu }{2}+1}}{\nu +2}\right )}+\frac {1}{x}}{2 a} \\
\end{align*}