60.7.181 problem 1772 (book 6.181)

Internal problem ID [11770]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1772 (book 6.181)
Date solved : Tuesday, January 28, 2025 at 06:11:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 29

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

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

✓ Solution by Mathematica

Time used: 0.895 (sec). Leaf size: 42

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

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