Internal problem ID [10298]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+1+\left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 27
dsolve((1+x^2)*diff(y(x),x$2)+1+diff(y(x),x)^2=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {x}{c_{1}}+\ln \left (x c_{1}-1\right )+\frac {\ln \left (x c_{1}-1\right )}{c_{1}^{2}}+c_{2} \]
✓ Solution by Mathematica
Time used: 7.046 (sec). Leaf size: 33
DSolve[(1+x^2)*y''[x]+1+(y'[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -x \cot (c_1)+\csc ^2(c_1) \log (-x \sin (c_1)-\cos (c_1))+c_2 \\ \end{align*}