62.34.2 problem Ex 2

Internal problem ID [12994]
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 58. Independent variable absent. Page 135
Problem number : Ex 2
Date solved : Tuesday, January 28, 2025 at 04:46:34 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y y^{\prime \prime }-{y^{\prime }}^{2}+1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.133 (sec). Leaf size: 59

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

\begin{align*} y &= -\frac {c_{1} \left ({\mathrm e}^{\frac {x +c_{2}}{c_{1}}}-{\mathrm e}^{\frac {-x -c_{2}}{c_{1}}}\right )}{2} \\ y &= \frac {c_{1} \left ({\mathrm e}^{\frac {x +c_{2}}{c_{1}}}-{\mathrm e}^{\frac {-x -c_{2}}{c_{1}}}\right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 14.308 (sec). Leaf size: 84

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

\begin{align*} y(x)\to -\frac {\sinh \left (\sqrt {e^{2 c_1}} (x+c_2)\right )}{\sqrt {e^{2 c_1}}} \\ y(x)\to \frac {\sinh \left (\sqrt {e^{2 c_1}} (x+c_2)\right )}{\sqrt {e^{2 c_1}}} \\ y(x)\to -x-c_2 \\ y(x)\to x+c_2 \\ \end{align*}