60.7.174 problem 1765 (book 6.174)

Internal problem ID [11763]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1765 (book 6.174)
Date solved : Tuesday, January 28, 2025 at 06:11:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.171 (sec). Leaf size: 22

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

\begin{align*} y &= 0 \\ y &= c_{1} \tanh \left (\frac {\ln \left (x \right )-c_{2}}{2 c_{1}}\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.054 (sec). Leaf size: 31

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

\[ \text {Solve}\left [\log (x)+\int _1^{y(x)}\frac {2}{-2 c_1 K[1]^2-1}dK[1]=c_2,y(x)\right ] \]