60.7.109 problem 1700 (book 6.109)

Internal problem ID [11698]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1700 (book 6.109)
Date solved : Monday, January 27, 2025 at 11:30:13 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime } y+{y^{\prime }}^{2}-y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.145 (sec). Leaf size: 34

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

\begin{align*} y &= 0 \\ y &= -c_{1} \left (\operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {-c_{1} -c_{2} -x}{c_{1}}}}{c_{1}}\right )+1\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.252 (sec). Leaf size: 90

DSolve[-D[y[x],x] + D[y[x],x]^2 + y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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