23.4.242 problem 242

Internal problem ID [6544]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 242
Date solved : Tuesday, September 30, 2025 at 03:04:03 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 118

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

\begin{align*} y &= -i \\ y &= i \\ y &= c_1 \\ y &= \frac {i c_1 -{\mathrm e}^{\frac {-4 \operatorname {LambertW}\left (-\frac {i {\mathrm e}^{\frac {\left (-x -c_2 +1\right ) c_1^{2}+\left (-2 x -2 c_2 -2\right ) c_1 -x -c_2 +1}{4 c_1}} \left (c_1 -1\right )}{4 c_1}\right ) c_1 +\left (-x -c_2 +1\right ) c_1^{2}+\left (-2 x -2 c_2 -2\right ) c_1 -x -c_2 +1}{4 c_1}}-i}{c_1 +1} \\ \end{align*}

✓ Mathematica. Time used: 30.092 (sec). Leaf size: 56

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

\begin{align*} y(x)&\to \csc (c_1) \sec (c_1) W\left (\sin (c_1) e^{-\left ((x+c_2) \cos ^2(c_1)\right )-\sin ^2(c_1)}\right )+\tan (c_1)\\ y(x)&\to e^{-x-c_2} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE (sqrt((27*y(x)**2*Derivative(y(x), (x, 2)) + 27*Derivative(y(x),