60.7.52 problem 1656 (book 6.65)

Internal problem ID [11602]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1656 (book 6.65)
Date solved : Sunday, March 30, 2025 at 08:31:05 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.346 (sec). Leaf size: 102

ode:=diff(diff(y(x),x),x)-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 \\ a \int _{}^{y}\frac {\textit {\_a}^{2}+2 c_1}{\sqrt {4-a^{2} \left (\textit {\_a}^{2}+2 c_1 \right )^{2}}}d \textit {\_a} -x -c_2 &= 0 \\ -a \int _{}^{y}\frac {\textit {\_a}^{2}+2 c_1}{\sqrt {4-a^{2} \left (\textit {\_a}^{2}+2 c_1 \right )^{2}}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 1.052 (sec). Leaf size: 1104

ode=-(a*y[x]*(1 + D[y[x],x]^2)^(3/2)) + 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(-a*(Derivative(y(x), x)**2 + 1)**(3/2)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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