23.4.259 problem 259
Internal
problem
ID
[6561]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
4.
THE
NONLINEAR
EQUATION
OF
SECOND
ORDER,
page
380
Problem
number
:
259
Date
solved
:
Friday, October 03, 2025 at 02:09:31 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (x +y\right ) \left (-y+x y^{\prime }\right )^{3}+x^{3} y^{2} y^{\prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.054 (sec). Leaf size: 170
ode:=(x+y(x))*(-y(x)+x*diff(y(x),x))^3+x^3*y(x)^2*diff(diff(y(x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {i \operatorname {BesselY}\left (i \sqrt {3}, 2 \sqrt {\textit {\_f}}\right ) \sqrt {3}\, 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*}
✓ Mathematica. Time used: 32.172 (sec). Leaf size: 248
ode=(x + y[x])*(-y[x] + x*D[y[x],x])^3 + x^3*y[x]^2*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [-\int _1^{\frac {y(x)}{x}}\frac {i \sqrt {3} \sqrt {K[2]} \operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )+\sqrt {K[2]} \operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )-2 \operatorname {BesselJ}\left (1+i \sqrt {3},2 \sqrt {K[2]}\right ) K[2]-2 \operatorname {BesselY}\left (1+i \sqrt {3},2 \sqrt {K[2]}\right ) c_1 K[2]+i \sqrt {3} \operatorname {BesselY}\left (i \sqrt {3},2 \sqrt {K[2]}\right ) c_1 \sqrt {K[2]}+\operatorname {BesselY}\left (i \sqrt {3},2 \sqrt {K[2]}\right ) c_1 \sqrt {K[2]}}{\left (\operatorname {BesselJ}\left (i \sqrt {3},2 \sqrt {K[2]}\right )+\operatorname {BesselY}\left (i \sqrt {3},2 \sqrt {K[2]}\right ) c_1\right ) K[2]^{3/2}}dK[2]-2 \log (x)+2 c_2=0,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**3*y(x)**2*Derivative(y(x), (x, 2)) + (x + y(x))*(x*Derivative(y(x), x) - y(x))**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -(-(x**3*y(x)**2*Derivative(y(x), (x, 2)) - x*y(x)**3 - y(x)**4)