60.7.89 problem 1680 (book 6.89)

Internal problem ID [11678]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1680 (book 6.89)
Date solved : Monday, January 27, 2025 at 11:29:53 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+{y^{\prime }}^{2}+1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 33

dsolve((x^2+1)*diff(diff(y(x),x),x)+diff(y(x),x)^2+1=0,y(x), singsol=all)
 

\[ y = \frac {\ln \left (c_{1} x -1\right ) c_{1}^{2}+c_{2} c_{1}^{2}+c_{1} x +\ln \left (c_{1} x -1\right )}{c_{1}^{2}} \]

✓ Solution by Mathematica

Time used: 0.958 (sec). Leaf size: 54

DSolve[1 + D[y[x],x]^2 + (1 + x^2)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+1}dK[1]\&\right ]\left [c_1+\int _1^{K[3]}-\frac {1}{K[2]^2+1}dK[2]\right ]dK[3]+c_2 \]