60.7.180 problem 1771 (book 6.180)

Internal problem ID [11769]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1771 (book 6.180)
Date solved : Tuesday, January 28, 2025 at 06:11:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

Time used: 0.806 (sec). Leaf size: 27

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

\begin{align*} y(x)\to 1+\frac {1}{c_2 x^2-c_1 x-1} \\ y(x)\to 1 \\ \end{align*}