60.8.6 problem 1842
Internal
problem
ID
[11767]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
7,
non-linear
third
and
higher
order
Problem
number
:
1842
Date
solved
:
Thursday, March 13, 2025 at 09:44:12 PM
CAS
classification
:
[[_3rd_order, _exact, _nonlinear], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2]]
\begin{align*} x^{2} y^{\prime \prime \prime }+x \left (-1+y\right ) y^{\prime \prime }+x {y^{\prime }}^{2}+\left (1-y\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.031 (sec). Leaf size: 190
ode:=x^2*diff(diff(diff(y(x),x),x),x)+x*(-1+y(x))*diff(diff(y(x),x),x)+x*diff(y(x),x)^2+(1-y(x))*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
\ln \left (x \right )+2 \left (\int _{}^{y}\frac {1}{2 \operatorname {RootOf}\left (-2 \sqrt {4+c_{1}}\, \operatorname {BesselY}\left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) c_{2} +2 \operatorname {BesselY}\left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) c_{2} \textit {\_h} +2 \operatorname {BesselY}\left (\frac {\sqrt {4+c_{1}}}{2}+1, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \sqrt {2}\, c_{2} \textit {\_Z} -4 \operatorname {BesselY}\left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) c_{2} +2 \operatorname {BesselJ}\left (\frac {\sqrt {4+c_{1}}}{2}+1, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \sqrt {2}\, \textit {\_Z} -2 \operatorname {BesselJ}\left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \sqrt {4+c_{1}}+2 \operatorname {BesselJ}\left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \textit {\_h} -4 \operatorname {BesselJ}\left (\frac {\sqrt {4+c_{1}}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right )\right )^{2}+\textit {\_h}^{2}-c_{1} -4 \textit {\_h}}d \textit {\_h} \right )-c_3 = 0
\]
✓ Mathematica. Time used: 60.15 (sec). Leaf size: 282
ode=(1 - y[x])*D[y[x],x] + x*D[y[x],x]^2 + x*(-1 + y[x])*D[y[x],{x,2}] + x^2*Derivative[3][y][x] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {2 \left (c_3 \left (\operatorname {BesselJ}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}},-\frac {1}{2} i x \sqrt {c_1}\right )-\frac {1}{4} i \sqrt {c_1} x \left (\operatorname {BesselJ}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}}-1,-\frac {1}{2} i x \sqrt {c_1}\right )-\operatorname {BesselJ}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}}+1,-\frac {1}{2} i x \sqrt {c_1}\right )\right )\right )+\operatorname {BesselY}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}},-\frac {1}{2} i x \sqrt {c_1}\right )-\frac {1}{4} i \sqrt {c_1} x \left (\operatorname {BesselY}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}}-1,-\frac {1}{2} i x \sqrt {c_1}\right )-\operatorname {BesselY}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}}+1,-\frac {1}{2} i x \sqrt {c_1}\right )\right )\right )}{c_3 \operatorname {BesselJ}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}},-\frac {1}{2} i x \sqrt {c_1}\right )+\operatorname {BesselY}\left (\frac {\sqrt {c_2+2}}{\sqrt {2}},-\frac {1}{2} i x \sqrt {c_1}\right )}
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), (x, 3)) + x*(y(x) - 1)*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x)**2 + (1 - y(x))*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (sqrt(-4*x**3*Derivative(y(x), (x, 3)) - 4*x**2*y(x)*Derivative(y(x), (x, 2)) + 4*x**2*Derivative(y(x), (x, 2)) + y(x)**2 - 2*y(x) + 1) + y(x) - 1)/(2*x) cannot be solved by the factorable group method