54.7.204 problem 1829 (book 6.238)

Internal problem ID [13053]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1829 (book 6.238)
Date solved : Friday, October 03, 2025 at 03:59:08 AM
CAS classification : [NONE]

\begin{align*} 2 \left (x^{2}+1\right ) {y^{\prime \prime }}^{2}-x y^{\prime \prime } \left (x +4 y^{\prime }\right )+2 \left (x +y^{\prime }\right ) y^{\prime }-2 y&=0 \end{align*}

✓ Maple. Time used: 0.268 (sec). Leaf size: 59

ode:=2*(x^2+1)*diff(diff(y(x),x),x)^2-x*diff(diff(y(x),x),x)*(x+4*diff(y(x),x))+2*(x+diff(y(x),x))*diff(y(x),x)-2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x \left (c_{1} +\frac {\operatorname {arcsinh}\left (x \right )}{4}\right ) \sqrt {x^{2}+1}}{2}-\frac {3 x^{2}}{16}+c_{1}^{2}+\frac {c_{1} \operatorname {arcsinh}\left (x \right )}{2}+\frac {\operatorname {arcsinh}\left (x \right )^{2}}{16} \\ y &= \frac {1}{2} c_{1} x^{2}+c_{2} x +c_{1}^{2}+c_{2}^{2} \\ \end{align*}

✓ Mathematica. Time used: 0.139 (sec). Leaf size: 32

ode=-2*y[x] + 2*D[y[x],x]*(x + D[y[x],x]) - x*(x + 4*D[y[x],x])*D[y[x],{x,2}] + 2*(1 + x^2)*D[y[x],{x,2}]^2 == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{2} \sqrt {c_2-c_1{}^2} x^2+c_1 x+c_2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(x + 4*Derivative(y(x), x))*Derivative(y(x), (x, 2)) + (2*x + 2*Derivative(y(x), x))*Derivative(y(x), x) + (2*x**2 + 2)*Derivative(y(x), (x, 2))**2 - 2*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x*(2*Derivative(y(x), (x, 2)) - 1)/2 - sqrt(-2*x**2*Derivative(y(x), (x, 2)) + x**2 + 4*y(x) - 4*Derivative(y(x), (x, 2))**2)/2 + Derivative(y(x), x) cannot be solved by the factorable group method