54.7.110 problem 1723 (book 6.132)

Internal problem ID [12959]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1723 (book 6.132)
Date solved : Wednesday, October 01, 2025 at 02:54:40 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.327 (sec). Leaf size: 115

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

\begin{align*} y &= -i x +c_1 \\ y &= i x +c_1 \\ \int _{}^{y}\frac {\textit {\_a}^{2} a +c_1}{\sqrt {-\textit {\_a}^{4} a^{2}-2 c_1 \,\textit {\_a}^{2} a -c_1^{2}+\textit {\_a}^{2}}}d \textit {\_a} -x -c_2 &= 0 \\ -\int _{}^{y}\frac {\textit {\_a}^{2} a +c_1}{\sqrt {-\textit {\_a}^{4} a^{2}-2 c_1 \,\textit {\_a}^{2} a -c_1^{2}+\textit {\_a}^{2}}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.993 (sec). Leaf size: 2181

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

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

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

Timed Out