62.38.6 problem Ex 6

Internal problem ID [12938]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 62. Summary. Page 144
Problem number : Ex 6
Date solved : Wednesday, March 05, 2025 at 08:54:14 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 19

ode:=y(x)*(1-ln(y(x)))*diff(diff(y(x),x),x)+(1+ln(y(x)))*diff(y(x),x)^2 = 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: 0.564 (sec). Leaf size: 159

ode=y[x]*(1-Log[y[x]])*D[y[x],{x,2}]+(1+Log[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)) + 1)) + Derivative(y(x), x) cannot be solved by the factorable group method