60.7.130 problem 1745 (book 6.154)

Internal problem ID [11680]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1745 (book 6.154)
Date solved : Sunday, March 30, 2025 at 08:41:50 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.115 (sec). Leaf size: 331

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

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

✓ Mathematica. Time used: 0.307 (sec). Leaf size: 51

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

\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (K[1]^2+1\right )}dK[1]\&\right ]\left [c_1+\frac {1}{2} \log (K[2])\right ]}dK[2]=x+c_2,y(x)\right ] \]

✗ Sympy

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

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