54.7.98 problem 1710 (book 6.119)

Internal problem ID [12947]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1710 (book 6.119)
Date solved : Wednesday, October 01, 2025 at 02:46:51 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime } y-{y^{\prime }}^{2}-\left (a y-1\right ) y^{\prime }+2 a^{2} y^{2}-2 b^{2} y^{3}+a y&=0 \end{align*}

✗ Maple

ode:=diff(diff(y(x),x),x)*y(x)-diff(y(x),x)^2-(-1+a*y(x))*diff(y(x),x)+2*a^2*y(x)^2-2*b^2*y(x)^3+a*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 105.793 (sec). Leaf size: 540

ode=a*y[x] + 2*a^2*y[x]^2 - 2*b^2*y[x]^3 - (-1 + a*y[x])*D[y[x],x] - D[y[x],x]^2 + y[x]*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{2 a}+e^{2 a x} \left (\frac {e^{-2 a x} \left (c_1 \left (a^{3/2}-\sqrt {a^3+2 b^2}\right ) \operatorname {Gamma}\left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )-2 c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}\right ) \sqrt {a b^2 c_2 e^{2 a x}} \operatorname {BesselJ}\left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )+\operatorname {Gamma}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}+1\right ) \left (\left (a^{3/2}+\sqrt {a^3+2 b^2}\right ) \operatorname {BesselJ}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )-2 \sqrt {a b^2 c_2 e^{2 a x}} \operatorname {BesselJ}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}+1,\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )\right )\right ){}^2}{4 a b^2 \left (c_1 \operatorname {Gamma}\left (1-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}\right ) \operatorname {BesselJ}\left (-\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )+\operatorname {Gamma}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}}+1\right ) \operatorname {BesselJ}\left (\frac {\sqrt {a^3+2 b^2}}{2 a^{3/2}},\frac {\sqrt {a b^2 e^{2 a x} c_2}}{a^{3/2}}\right )\right ){}^2}+c_2\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(2*a**2*y(x)**2 + a*y(x) - 2*b**2*y(x)**3 - (a*y(x) - 1)*Derivative(y(x), x) + y(x)*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE a*y(x)/2 - sqrt(9*a**2*y(x)**2 + 2*a*y(x) - 8*b**2*y(x)**3 + 4*y(x)*Derivative(y(x), (x, 2)) + 1)/2 + Derivative(y(x), x) - 1/2 cannot be solved by the factorable group method