54.8.6 problem 1842

Internal problem ID [13066]
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 : Friday, October 03, 2025 at 03:59:11 AM
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: 0.032 (sec). Leaf size: 190

ode:=x^2*diff(diff(diff(y(x),x),x),x)+x*(y(x)-1)*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);
 

\[ \ln \left (x \right )+2 \int _{}^{y}\frac {1}{2 \operatorname {RootOf}\left (-2 \operatorname {BesselY}\left (\frac {\sqrt {4+c_1}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \sqrt {4+c_1}\, c_2 +2 \operatorname {BesselY}\left (\frac {\sqrt {4+c_1}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) c_2 \textit {\_h} -4 \operatorname {BesselY}\left (\frac {\sqrt {4+c_1}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) c_2 +2 \operatorname {BesselY}\left (\frac {\sqrt {4+c_1}}{2}+1, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \sqrt {2}\, c_2 \textit {\_Z} +2 \operatorname {BesselJ}\left (\frac {\sqrt {4+c_1}}{2}+1, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \sqrt {2}\, \textit {\_Z} -2 \operatorname {BesselJ}\left (\frac {\sqrt {4+c_1}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \sqrt {4+c_1}+2 \operatorname {BesselJ}\left (\frac {\sqrt {4+c_1}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right ) \textit {\_h} -4 \operatorname {BesselJ}\left (\frac {\sqrt {4+c_1}}{2}, \frac {\sqrt {2}\, \textit {\_Z}}{2}\right )\right )^{2}+\textit {\_h}^{2}-c_1 -4 \textit {\_h}}d \textit {\_h} -c_3 = 0 \]

✗ Mathematica

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]
 

Not solved

✗ 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