62.33.1 problem Ex 1

Internal problem ID [12988]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 57. Dependent variable absent. Page 132
Problem number : Ex 1
Date solved : Tuesday, January 28, 2025 at 04:46:25 AM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Solution by Maple

Time used: 0.025 (sec). Leaf size: 33

dsolve((1+x^2)*diff(y(x),x$2)+1+diff(y(x),x)^2=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: 0.912 (sec). Leaf size: 54

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

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