Internal problem ID [9977]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1655 (book 6.64).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]
\[ \boxed {y^{\prime \prime }-2 a x \left ({y^{\prime }}^{2}+1\right )^{\frac {3}{2}}=0} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 57
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 \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= a \left (\int \sqrt {-\frac {1}{-1+\left (x^{2}+2 c_{1} \right )^{2} a^{2}}}\, \left (x^{2}+2 c_{1} \right )d x \right )+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 60.462 (sec). Leaf size: 308
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 {\frac {a x^2-1+c_1}{-1+c_1}} \sqrt {\frac {a x^2+1+c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{1+c_1}} \sqrt {a^2 x^4+2 a c_1 x^2-1+c_1{}^2}} \\ y(x)\to \frac {\sqrt {\frac {a x^2-1+c_1}{-1+c_1}} \sqrt {\frac {a x^2+1+c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{c_1+1}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{1+c_1}} \sqrt {a^2 x^4+2 a c_1 x^2-1+c_1{}^2}}+c_2 \\ \end{align*}