60.7.169 problem 1760 (book 6.169)

Internal problem ID [11758]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1760 (book 6.169)
Date solved : Monday, January 27, 2025 at 11:34:15 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 35

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

\begin{align*} y &= 0 \\ y &= \sqrt {c_{1} x^{2}+2 c_{2}} \\ y &= -\sqrt {c_{1} x^{2}+2 c_{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.255 (sec). Leaf size: 18

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

\[ y(x)\to c_2 \sqrt {x^2+c_1} \]