60.7.119 problem 1710 (book 6.119)
Internal
problem
ID
[11708]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1710
(book
6.119)
Date
solved
:
Monday, January 27, 2025 at 11:30:35 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} y^{\prime \prime } y-{y^{\prime }}^{2}-\left (-1+a y\right ) y^{\prime }+2 a^{2} y^{2}-2 b^{2} y^{3}+a y&=0 \end{align*}
✗ Solution by Maple
dsolve(diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2-(a*y(x)-1)*diff(y(x),x)+2*a^2*y(x)^2-2*b^2*y(x)^3+a*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 107.868 (sec). Leaf size: 540
DSolve[a*y[x] + 2*a^2*y[x]^2 - 2*b^2*y[x]^3 - (-1 + a*y[x])*D[y[x],x] - D[y[x],x]^2 + y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to -\frac {1}{2 a}+e^{2 a x} \left (\frac {e^{-2 a x} \left (c_1 \left (a^{3/2}-\sqrt {a^3+2 b^2}\right ) \operatorname {Gamma}\left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )-2 c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}\right ) \sqrt {a b^2 c_2 e^{2 a x}} \operatorname {BesselJ}\left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )+\operatorname {Gamma}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}+1\right ) \left (\left (a^{3/2}+\sqrt {a^3+2 b^2}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )-2 \sqrt {a b^2 c_2 e^{2 a x}} \operatorname {BesselJ}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}+1,\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )\right )\right ){}^2}{4 a b^2 \left (c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )+\operatorname {Gamma}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}+1\right ) \operatorname {BesselJ}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )\right ){}^2}+c_2\right )
\]