60.7.116 problem 1729 (book 6.138)

Internal problem ID [11666]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1729 (book 6.138)
Date solved : Sunday, March 30, 2025 at 08:37:33 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 24

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

\[ y = \frac {\left (c_{1}^{2}-a \right ) x^{2}}{4 c_{2}}+c_{1} x +c_{2} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 36

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

\begin{align*} y(x)\to \frac {x^2 \left (-a+c_1{}^2\right )}{4 c_2}+c_1 x+c_2 \\ y(x)\to \text {Indeterminate} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a + 2*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 -sqrt(a + 2*y(x)*Derivative(y(x), (x, 2))) + Derivative(y(x), x) cannot be solved by the factorable group method