60.7.222 problem 1813 (book 6.222)

Internal problem ID [11811]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1813 (book 6.222)
Date solved : Monday, January 27, 2025 at 11:37:00 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*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 19

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

\[ y = {\mathrm e}^{\frac {c_{1} x +c_{2} -1}{c_{1} x +c_{2}}} \]

✓ Solution by Mathematica

Time used: 0.528 (sec). Leaf size: 159

DSolve[(1 + Log[y[x]])*D[y[x],x]^2 + (1 - Log[y[x]])*y[x]*D[y[x],{x,2}] == 0,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*}