60.7.163 problem 1781 (book 6.190)

Internal problem ID [11713]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1781 (book 6.190)
Date solved : Sunday, March 30, 2025 at 08:44:25 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{2} y^{\prime \prime }+y {y^{\prime }}^{2}-a x -b&=0 \end{align*}

✓ Maple. Time used: 0.107 (sec). Leaf size: 171

ode:=y(x)^2*diff(diff(y(x),x),x)+y(x)*diff(y(x),x)^2-a*x-b = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (\sqrt {3}\, b \int _{}^{\textit {\_Z}}-\frac {\textit {\_g}^{2} \left (-\sqrt {3}\, \left (-\frac {a}{\textit {\_g}^{3} b^{3}}\right )^{{1}/{3}} b +2 \sqrt {3}\, a -3 b \left (-\frac {a}{\textit {\_g}^{3} b^{3}}\right )^{{1}/{3}} \tan \left (\operatorname {RootOf}\left (-2 b^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3} a^{2}-1\right )}{\sum }\frac {\ln \left (\textit {\_g} -\textit {\_R} \right )}{\textit {\_R}^{2}}\right ) \left (-\frac {a}{\textit {\_g}^{3} b^{3}}\right )^{{2}/{3}} \textit {\_g}^{2}+2 \textit {\_Z} \sqrt {3}\, a^{2}-\ln \left (\frac {1}{\sqrt {3}\, \sin \left (2 \textit {\_Z} \right )+2+\cos \left (2 \textit {\_Z} \right )}\right ) a^{2}-6 c_1 \,a^{2}\right )\right )\right )}{\textit {\_g}^{3} a^{2}-1}d \textit {\_g} a -6 b \ln \left (a x +b \right )+6 c_2 a \right ) \left (a x +b \right ) \]

✗ Mathematica

ode=-b - a*x + y[x]*D[y[x],x]^2 + y[x]^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-a*x - b + y(x)**2*Derivative(y(x), (x, 2)) + y(x)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt((a*x + b - y(x)**2*Derivative(y(x), (x, 2)))/y(x)) + Derivative(y(x), x) cannot be solved by the factorable group method