32.10.3 problem Exercise 35.3, page 504

Internal problem ID [5997]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 8. Special second order equations. Lesson 35. Independent variable x absent
Problem number : Exercise 35.3, page 504
Date solved : Monday, January 27, 2025 at 01:30:38 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Solution by Maple

Time used: 0.067 (sec). Leaf size: 59

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

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

✓ Solution by Mathematica

Time used: 13.158 (sec). Leaf size: 84

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