Internal problem ID [9399]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1825 (book 6.234).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_y]]
\[ \boxed {\left (a \sqrt {{y^{\prime }}^{2}+1}-y^{\prime } x \right ) y^{\prime \prime }-{y^{\prime }}^{2}-1=0} \]
✓ Solution by Maple
Time used: 0.328 (sec). Leaf size: 117
dsolve((a*(diff(y(x),x)^2+1)^(1/2)-x*diff(y(x),x))*diff(diff(y(x),x),x)-diff(y(x),x)^2-1=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = -i x +c_{1} \\ y \left (x \right ) = i x +c_{1} \\ y \left (x \right ) = \int \frac {-c_{1} a^{2}+x \sqrt {a^{2} \left (c_{1}^{2}+a^{2}-x^{2}\right )}}{a \left (a^{2}-x^{2}\right )}d x +c_{2} \\ y \left (x \right ) = \int -\frac {c_{1} a^{2}+x \sqrt {a^{2} \left (c_{1}^{2}+a^{2}-x^{2}\right )}}{a \left (a^{2}-x^{2}\right )}d x +c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.638 (sec). Leaf size: 326
DSolve[-1 - y'[x]^2 + (-(x*y'[x]) + a*Sqrt[1 + y'[x]^2])*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )} \left (-2 \sqrt {-a^2+x^2-c_1{}^2}-\left (c_1 \left (\cot ^{-1}\left (\frac {c_1 \sqrt {-a^2+x^2-c_1{}^2}}{a^2-a x+c_1{}^2}\right )+\cot ^{-1}\left (\frac {c_1 \sqrt {-a^2+x^2-c_1{}^2}}{a (a+x)+c_1{}^2}\right )\right )\right )\right )}{2 x \sqrt {-a^2+x^2-c_1{}^2}}+c_1 \left (-\text {arctanh}\left (\frac {x}{a}\right )\right )+c_2 \\ y(x)\to \frac {\sqrt {x^2 \left (a^2-x^2+c_1{}^2\right )} \left (2 \sqrt {-a^2+x^2-c_1{}^2}+c_1 \left (\cot ^{-1}\left (\frac {c_1 \sqrt {-a^2+x^2-c_1{}^2}}{a^2-a x+c_1{}^2}\right )+\cot ^{-1}\left (\frac {c_1 \sqrt {-a^2+x^2-c_1{}^2}}{a (a+x)+c_1{}^2}\right )\right )\right )}{2 x \sqrt {-a^2+x^2-c_1{}^2}}+c_1 \left (-\text {arctanh}\left (\frac {x}{a}\right )\right )+c_2 \\ \end{align*}