Internal problem ID [6093]
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: 28.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-1-\left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.052 (sec). Leaf size: 17
dsolve(diff(y(x),x$2)=1+diff(y(x),x)^2,y(x), singsol=all)
\[ y \relax (x ) = -\ln \left (c_{1} \sin \relax (x )-c_{2} \cos \relax (x )\right ) \]
✓ Solution by Mathematica
Time used: 2.057 (sec). Leaf size: 16
DSolve[y''[x]==1+(y'[x])^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-\log (\cos (x+c_1)) \\ \end{align*}