60.7.192 problem 1783 (book 6.192)

Internal problem ID [11781]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1783 (book 6.192)
Date solved : Monday, January 27, 2025 at 11:35:43 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} \left (1+y^{2}\right ) y^{\prime \prime }-3 y {y^{\prime }}^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.135 (sec). Leaf size: 41

dsolve((y(x)^2+1)*diff(diff(y(x),x),x)-3*y(x)*diff(y(x),x)^2=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.020 (sec). Leaf size: 173

DSolve[-3*y[x]*D[y[x],x]^2 + (1 + y[x]^2)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {i c_1 (x+c_2)}{\sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x-1+c_2{}^2 c_1{}^2}} \\ y(x)\to \frac {i c_1 (x+c_2)}{\sqrt {c_1{}^2 x^2+2 c_2 c_1{}^2 x-1+c_2{}^2 c_1{}^2}} \\ y(x)\to -\frac {i c_1}{\sqrt {c_1{}^2}} \\ y(x)\to \frac {i c_1}{\sqrt {c_1{}^2}} \\ y(x)\to -\frac {i (x+c_2)}{\sqrt {(x+c_2){}^2}} \\ y(x)\to \frac {i (x+c_2)}{\sqrt {(x+c_2){}^2}} \\ \end{align*}