Internal problem ID [9234]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1655.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-a y \left (\left (y^{\prime }\right )^{2}+1\right )^{\frac {3}{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.081 (sec). Leaf size: 124
dsolve(diff(diff(y(x),x),x)-a*y(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} \\ \int _{}^{y \relax (x )}\frac {\left (\textit {\_a}^{2}+2 c_{1}\right ) a}{\sqrt {-a^{2} \textit {\_a}^{4}-4 \textit {\_a}^{2} a^{2} c_{1}-4 a^{2} c_{1}^{2}+4}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {\left (\textit {\_a}^{2}+2 c_{1}\right ) a}{\sqrt {-a^{2} \textit {\_a}^{4}-4 \textit {\_a}^{2} a^{2} c_{1}-4 a^{2} c_{1}^{2}+4}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.529 (sec). Leaf size: 350
DSolve[-(a*y[x]*(1 + y'[x]^2)^(3/2)) + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 c_1}{-1+c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 c_1}{1+c_1}} \left (F\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1-4+4 c_1{}^2}}\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\frac {\sqrt {\frac {\text {$\#$1}^2 a-2+2 c_1}{-1+c_1}} \sqrt {\frac {\text {$\#$1}^2 a+2+2 c_1}{1+c_1}} \left (F\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{2 c_1+2}} \text {$\#$1}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {\text {$\#$1}^4 a^2+4 \text {$\#$1}^2 a c_1-4+4 c_1{}^2}}\&\right ][x+c_2] \\ \end{align*}