60.7.221 problem 1812 (book 6.221)

Internal problem ID [11810]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1812 (book 6.221)
Date solved : Tuesday, January 28, 2025 at 06:11:27 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.312 (sec). Leaf size: 85

dsolve((y(x)^2+x^2)^(1/2)*diff(diff(y(x),x),x)-a*(diff(y(x),x)^2+1)^(3/2)=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 {\operatorname {RootOf}\left (\arctan \left (\textit {\_g} \right )+\int _{}^{\textit {\_Z}}\frac {1+\sqrt {a^{2} \left (\textit {\_f}^{2}+1\right )}}{\left (\textit {\_f}^{2} a^{2}+a^{2}-1\right ) \left (\textit {\_f}^{2}+1\right )}d \textit {\_f} +c_{1} \right )-\textit {\_g}}{\textit {\_g}^{2}+1}d \textit {\_g} +c_{2} \right ) x \\ \end{align*}

✗ Solution by Mathematica

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

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

Timed out