23.4.276 problem 276

Internal problem ID [6578]
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 : 276
Date solved : Tuesday, September 30, 2025 at 03:10:58 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 19

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

\[ y = {\mathrm e}^{\frac {c_1 x +c_2 -1}{c_1 x +c_2}} \]

✓ Mathematica. Time used: 0.343 (sec). Leaf size: 159

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[2]}\frac {\log (K[1])+1}{K[1] (\log (K[1])-1)}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[2]}\frac {\log (K[1])+1}{K[1] (\log (K[1])-1)}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[2]}\frac {\log (K[1])+1}{K[1] (\log (K[1])-1)}dK[1]\right )}{c_1}dK[2]\&\right ][x+c_2] \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt((log(y(x)) - 1)*y(x)*Derivative(y(x), (x, 2))/(log(y(x)) +