56.3.23 problem 23

Internal problem ID [8881]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 23
Date solved : Monday, January 27, 2025 at 05:18:25 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 22

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

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

✓ Solution by Mathematica

Time used: 60.266 (sec). Leaf size: 33

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

\[ y(x)\to \int _1^x-\frac {2}{2 c_1-\log \left (K[1]^2+1\right )}dK[1]+c_2 \]