Internal problem ID [9258]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1679.
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 {\left (x^{2}+1\right ) y^{\prime \prime }+\left (y^{\prime }\right )^{2}+1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.02 (sec). Leaf size: 27
dsolve((x^2+1)*diff(diff(y(x),x),x)+diff(y(x),x)^2+1=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {x}{c_{1}}+\ln \left (x c_{1}-1\right )+\frac {\ln \left (x c_{1}-1\right )}{c_{1}^{2}}+c_{2} \]
✓ Solution by Mathematica
Time used: 7.902 (sec). Leaf size: 33
DSolve[1 + y'[x]^2 + (1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x \cot (c_1)+\csc ^2(c_1) \log (-x \sin (c_1)-\cos (c_1))+c_2 \\ \end{align*}