60.7.208 problem 1799 (book 6.208)
Internal
problem
ID
[11797]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1799
(book
6.208)
Date
solved
:
Tuesday, January 28, 2025 at 06:11:26 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{3} y^{2} y^{\prime \prime }+\left (x +y\right ) \left (-y+x y^{\prime }\right )^{3}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.151 (sec). Leaf size: 160
dsolve(x^3*y(x)^2*diff(diff(y(x),x),x)+(x+y(x))*(x*diff(y(x),x)-y(x))^3=0,y(x), singsol=all)
\begin{align*}
y &= 0 \\
y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {i \sqrt {3}\, \operatorname {BesselY}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) c_{1} \sqrt {\textit {\_f}}+i \sqrt {3}\, \operatorname {BesselJ}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \sqrt {\textit {\_f}}+\operatorname {BesselY}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) c_{1} \sqrt {\textit {\_f}}-2 c_{1} \operatorname {BesselY}\left (1+i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \textit {\_f} +\operatorname {BesselJ}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \sqrt {\textit {\_f}}-2 \operatorname {BesselJ}\left (1+i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \textit {\_f}}{\textit {\_f}^{{3}/{2}} \left (\operatorname {BesselY}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) c_{1} +\operatorname {BesselJ}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right )\right )}d \textit {\_f} +2 c_{2} \right ) x \\
\end{align*}
✓ Solution by Mathematica
Time used: 33.346 (sec). Leaf size: 248
DSolve[(x + y[x])*(-y[x] + x*D[y[x],x])^3 + x^3*y[x]^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [-\int _1^{\frac {y(x)}{x}}\frac {i \sqrt {3} \sqrt {K[2]} \operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )+\sqrt {K[2]} \operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )-2 \operatorname {BesselJ}\left (1+i \sqrt {3},2 \sqrt {K[2]}\right ) K[2]-2 \operatorname {BesselY}\left (1+i \sqrt {3},2 \sqrt {K[2]}\right ) c_1 K[2]+i \sqrt {3} \operatorname {BesselY}\left (i \sqrt {3},2 \sqrt {K[2]}\right ) c_1 \sqrt {K[2]}+\operatorname {BesselY}\left (i \sqrt {3},2 \sqrt {K[2]}\right ) c_1 \sqrt {K[2]}}{\left (\operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )+\operatorname {BesselY}\left (i \sqrt {3},2 \sqrt {K[2]}\right ) c_1\right ) K[2]^{3/2}}dK[2]-2 \log (x)+2 c_2=0,y(x)\right ]
\]