Internal problem ID [4145]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {y y^{\prime \prime }-\left (y^{\prime }\right )^{2}+1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.077 (sec). Leaf size: 79
dsolve(y(x)*diff(y(x),x$2)=(diff(y(x),x))^2-1,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {c_{1} \left ({\mathrm e}^{-\frac {2 c_{2}}{c_{1}}} {\mathrm e}^{-\frac {2 x}{c_{1}}}-1\right ) {\mathrm e}^{\frac {c_{2}}{c_{1}}} {\mathrm e}^{\frac {x}{c_{1}}}}{2} \\ y \relax (x ) = \frac {c_{1} \left ({\mathrm e}^{\frac {2 c_{2}}{c_{1}}} {\mathrm e}^{\frac {2 x}{c_{1}}}-1\right ) {\mathrm e}^{-\frac {c_{2}}{c_{1}}} {\mathrm e}^{-\frac {x}{c_{1}}}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.2 (sec). Leaf size: 79
DSolve[y[x]*y''[x]==(y'[x])^2-1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {1}{2} e^{-e^{c_1} (x+c_2)-c_1} \left (-1+e^{2 e^{c_1} (x+c_2)}\right ) \\ y(x)\to \frac {1}{2} e^{-e^{c_1} (x+c_2)-c_1} \left (-1+e^{2 e^{c_1} (x+c_2)}\right ) \\ \end{align*}