60.7.179 problem 1770 (book 6.179)

Internal problem ID [11768]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1770 (book 6.179)
Date solved : Monday, January 27, 2025 at 11:34:23 PM
CAS classification : [_Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.059 (sec). Leaf size: 25

dsolve(2*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 &= c_{1} \sqrt {x}\, c_{2} +c_{1}^{2} x +\frac {c_{2}^{2}}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.237 (sec). Leaf size: 18

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

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