66.2.9 problem Problem 9

Internal problem ID [13908]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 9
Date solved : Tuesday, January 28, 2025 at 06:08:14 AM
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: 0.024 (sec). Leaf size: 33

dsolve((1+x^2)*diff(y(x),x$2)+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: 6.418 (sec). Leaf size: 33

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

\[ y(x)\to -x \cot (c_1)+\csc ^2(c_1) \log (-x \sin (c_1)-\cos (c_1))+c_2 \]