60.7.173 problem 1764 (book 6.173)

Internal problem ID [11762]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1764 (book 6.173)
Date solved : Monday, January 27, 2025 at 11:34:18 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: 0.056 (sec). Leaf size: 162

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 {3^{{1}/{3}} \left (\left (a -1\right )^{2} \left (-x^{2 a +1} c_{1} +c_{2} x^{3 a} \left (a -1\right )\right )\right )^{{1}/{3}} x^{-a}}{a -1} \\ y &= -\frac {\left (\left (a -1\right )^{2} \left (-x^{2 a +1} c_{1} +c_{2} x^{3 a} \left (a -1\right )\right )\right )^{{1}/{3}} \left (3^{{1}/{3}}+i 3^{{5}/{6}}\right ) x^{-a}}{2 a -2} \\ y &= \frac {\left (\left (a -1\right )^{2} \left (-x^{2 a +1} c_{1} +c_{2} x^{3 a} \left (a -1\right )\right )\right )^{{1}/{3}} \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) x^{-a}}{2 a -2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 3.799 (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]
 

\[ y(x)\to c_2 \sqrt [3]{3 x^{1-a}-a c_1+c_1} \]