Internal problem ID [6073]
Book: Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam
Publishing Co. NY. 6th edition. 1981.
Section: CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES
Page 324
Problem number: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {\left (y+1\right ) y^{\prime \prime }-\left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.089 (sec). Leaf size: 16
dsolve((y(x)+1)*diff(y(x),x$2)=diff(y(x),x)^2,y(x), singsol=all)
\begin{align*} y \relax (x ) = -1 \\ y \relax (x ) = {\mathrm e}^{c_{1} x} c_{2}-1 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.094 (sec). Leaf size: 21
DSolve[(y[x]+1)*y''[x]==(y'[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -1+\frac {c_1{}^{x+\log (c_2)}}{\log (c_1)} \\ \end{align*}