Internal problem ID [9976]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1654 (book 6.63).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {y^{\prime \prime }-a \left ({y^{\prime }}^{2}+1\right )^{\frac {3}{2}}=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 59
dsolve(diff(diff(y(x),x),x)=a*(diff(y(x),x)^2+1)^(3/2),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 ) &= \frac {\left (-1+\left (c_{1} +x \right )^{2} a^{2}\right ) \sqrt {-\frac {1}{-1+\left (c_{1} +x \right )^{2} a^{2}}}+c_{2} a}{a} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.835 (sec). Leaf size: 75
DSolve[-(a*(1 + y'[x]^2)^(3/2)) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-\frac {i \sqrt {a^2 x^2+2 a c_1 x-1+c_1{}^2}}{a} \\ y(x)\to \frac {i \sqrt {a^2 x^2+2 a c_1 x-1+c_1{}^2}}{a}+c_2 \\ \end{align*}