60.7.176 problem 1767 (book 6.176)

Internal problem ID [11765]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1767 (book 6.176)
Date solved : Monday, January 27, 2025 at 11:34:20 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 }-4 x {y^{\prime }}^{2}+4 y^{\prime } y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.054 (sec). Leaf size: 66

dsolve(x*y(x)*diff(diff(y(x),x),x)-4*x*diff(y(x),x)^2+4*y(x)*diff(y(x),x)=0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= \frac {x}{\left (-3 c_{2} x^{3}+c_{1} \right )^{{1}/{3}}} \\ y &= -\frac {\left (1+i \sqrt {3}\right ) x}{2 \left (-3 c_{2} x^{3}+c_{1} \right )^{{1}/{3}}} \\ y &= \frac {\left (i \sqrt {3}-1\right ) x}{2 \left (-3 c_{2} x^{3}+c_{1} \right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 7.143 (sec). Leaf size: 43

DSolve[4*y[x]*D[y[x],x] - 4*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 c_2 \exp \left (\int _1^x\frac {1}{c_1 K[1]^4+K[1]}dK[1]\right ) \\ y(x)\to c_2 \\ y(x)\to c_2 x \\ \end{align*}