Internal problem ID [9233]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1654.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-2 a x \left (\left (y^{\prime }\right )^{2}+1\right )^{\frac {3}{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.068 (sec). Leaf size: 67
dsolve(diff(diff(y(x),x),x)-2*a*x*(diff(y(x),x)^2+1)^(3/2)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -i x +c_{1} \\ y \relax (x ) = i x +c_{1} \\ y \relax (x ) = \int \sqrt {-\frac {1}{a^{2} x^{4}+4 a^{2} c_{1} x^{2}+4 a^{2} c_{1}^{2}-1}}\, a \left (x^{2}+2 c_{1}\right )d x +c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.25 (sec). Leaf size: 288
DSolve[-2*a*x*(1 + y'[x]^2)^(3/2) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-\frac {\sqrt {1+\frac {a x^2}{-1+c_1}} \sqrt {1+\frac {a x^2}{1+c_1}} \left (F\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{c_1+1}}\right )|1+\frac {2}{c_1-1}\right )+(-1+c_1) E\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{c_1+1}}\right )|1+\frac {2}{c_1-1}\right )\right )}{\sqrt {\frac {a}{1+c_1}} \sqrt {\left (a x^2-1+c_1\right ) \left (a x^2+1+c_1\right )}} \\ y(x)\to \frac {\sqrt {1+\frac {a x^2}{-1+c_1}} \sqrt {1+\frac {a x^2}{1+c_1}} \left (F\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{c_1+1}}\right )|1+\frac {2}{c_1-1}\right )+(-1+c_1) E\left (i \sinh ^{-1}\left (x \sqrt {\frac {a}{c_1+1}}\right )|1+\frac {2}{c_1-1}\right )\right )}{\sqrt {\frac {a}{1+c_1}} \sqrt {\left (a x^2-1+c_1\right ) \left (a x^2+1+c_1\right )}}+c_2 \\ \end{align*}