54.7.50 problem 1654 (book 6.63)

Internal problem ID [12899]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1654 (book 6.63)
Date solved : Wednesday, October 01, 2025 at 02:45:53 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.255 (sec). Leaf size: 57

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

\begin{align*} y &= -i x +c_1 \\ y &= i x +c_1 \\ y &= \frac {\left (-1+a^{2} \left (c_1 +x \right )^{2}\right ) \sqrt {-\frac {1}{-1+a^{2} \left (c_1 +x \right )^{2}}}+c_2 a}{a} \\ \end{align*}

✓ Mathematica. Time used: 0.386 (sec). Leaf size: 75

ode=-(a*(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*} y(x)&\to c_2-\frac {i \sqrt {a^2 x^2+2 a c_1 x-1+c_1{}^2}}{a}\\ y(x)&\to \frac {i \sqrt {a^2 x^2+2 a c_1 x-1+c_1{}^2}}{a}+c_2 \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) + 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)**(1/3)/2 + sqrt(3)*I*(Derivative(y(x), (x, 2))**2/a**2)**(1/3)/2 - 1) + Derivative(y(x), x) cannot be solved by the factorable group method