Internal problem ID [6847]
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: 30.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {y^{\prime \prime }-\left (1+{y^{\prime }}^{2}\right )^{\frac {3}{2}}=0} \]
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 49
dsolve(diff(y(x),x$2)=(1+diff(y(x),x)^2)^(3/2),y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -i x +c_{1} \\ y \left (x \right ) &= i x +c_{1} \\ y \left (x \right ) &= \left (c_{1} +x +1\right ) \left (x -1+c_{1} \right ) \sqrt {-\frac {1}{\left (c_{1} +x +1\right ) \left (x -1+c_{1} \right )}}+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.269 (sec). Leaf size: 59
DSolve[y''[x]==(1+(y'[x])^2)^(3/2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-i \sqrt {x^2+2 c_1 x-1+c_1{}^2} \\ y(x)\to i \sqrt {x^2+2 c_1 x-1+c_1{}^2}+c_2 \\ \end{align*}