54.7.113 problem 1726 (book 6.135)

Internal problem ID [12962]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1726 (book 6.135)
Date solved : Friday, October 03, 2025 at 03:58:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.104 (sec). Leaf size: 107

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

\begin{align*} y &= x +\operatorname {RootOf}\left (-x -\int _{}^{\textit {\_Z}}\frac {c_1^{2} \textit {\_f}^{2}-1}{c_1^{2} \textit {\_f}^{2}+\sqrt {-c_1^{2} \textit {\_f}^{2}+2}\, c_1 \textit {\_f} -2}d \textit {\_f} +c_2 \right ) \\ y &= x +\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}-\frac {c_1^{2} \textit {\_f}^{2}-1}{-2+c_1^{2} \textit {\_f}^{2}-\sqrt {-c_1^{2} \textit {\_f}^{2}+2}\, c_1 \textit {\_f}}d \textit {\_f} +c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.157 (sec). Leaf size: 109

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

\[ \text {Solve}\left [\left \{x=\int \frac {\exp \left (-\int _1^{K[4]}\frac {K[3]-1}{K[3]^3+K[3]^2+K[3]+1}dK[3]-c_1\right )}{K[4]^3+K[4]^2+K[4]+1} \, dK[4]+c_2,y(x)=x-\exp \left (-\int _1^{K[4]}\frac {K[3]-1}{K[3]^3+K[3]^2+K[3]+1}dK[3]-c_1\right )\right \},\{y(x),K[4]\}\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE (-27*x*Derivative(y(x), (x, 2))/2 + sqrt((-27*x*Derivative(y(x), (x, 2)) + 27*y(x)*Derivative(y(x), (x, 2)) + 20)**2 + 32)/2 + 27*y(x)*Derivative(y(x), (x, 2))/2 + 10)**(1/3)/3 + Derivative(y(x), x) + 1/3 - 2/(3*(-27*x*Derivative(y(x), (x, 2))/2 + sqrt((-27*x*Derivative(y(x), (x, 2)) + 27*y(x)*Derivative(y(x), (x, 2)) + 20)**2 + 32)/2 + 27*y(x)*Derivative(y(x), (x, 2))/2 + 10)**(1/3)) cannot be solved by the factorable group method