60.7.175 problem 1766 (book 6.175)

Internal problem ID [11764]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1766 (book 6.175)
Date solved : Monday, January 27, 2025 at 11:34:19 PM
CAS classification : [_Liouville, [_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}+a y y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.060 (sec). Leaf size: 31

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

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

✓ Solution by Mathematica

Time used: 0.509 (sec). Leaf size: 29

DSolve[a*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]
 

\begin{align*} y(x)\to \frac {c_2 x^a}{x+(a-1) c_1 x^a} \\ y(x)\to 0 \\ \end{align*}