60.7.124 problem 1715 (book 6.124)

Internal problem ID [11713]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1715 (book 6.124)
Date solved : Monday, January 27, 2025 at 11:30:55 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime } y-3 {y^{\prime }}^{2}+3 y^{\prime } y-y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.073 (sec). Leaf size: 72

dsolve(diff(diff(y(x),x),x)*y(x)-3*diff(y(x),x)^2+3*y(x)*diff(y(x),x)-y(x)^2=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.464 (sec). Leaf size: 47

DSolve[-y[x]^2 + 3*y[x]*D[y[x],x] - 3*D[y[x],x]^2 + y[x]*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 \exp \left (\int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) (2 K[1]-1)}dK[1]\&\right ][c_1+K[2]]dK[2]\right ) \]