81.7.7 problem 8

Internal problem ID [18704]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter VI. Certain particular forms of equations. Exercises at page 74
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 12:12:03 PM
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: 0.011 (sec). Leaf size: 37

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

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

✓ Solution by Mathematica

Time used: 7.844 (sec). Leaf size: 33

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

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