60.7.228 problem 1819 (book 6.228)

Internal problem ID [11817]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1819 (book 6.228)
Date solved : Tuesday, January 28, 2025 at 06:11:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.174 (sec). Leaf size: 78

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

\begin{align*} y &= -i x \\ y &= i x \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {-\textit {\_f} +\operatorname {RootOf}\left (-\tan \left (\frac {1}{\textit {\_Z}}\right ) c_{1} \textit {\_Z} +\textit {\_f} c_{1} \tan \left (\frac {1}{\textit {\_Z}}\right )+\tan \left (\frac {1}{\textit {\_Z}}\right ) \textit {\_Z} \textit {\_f} +c_{1} \textit {\_f} \textit {\_Z} +\tan \left (\frac {1}{\textit {\_Z}}\right )+c_{1} +\textit {\_Z} -\textit {\_f} \right )}{\textit {\_f}^{2}+1}d \textit {\_f} +c_{2} \right ) x \\ \end{align*}

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

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

Not solved