Internal problem ID [6096]
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: 33.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {\left (y y^{\prime \prime }+1+{y^{\prime }}^{2}\right )^{2}-\left (1+{y^{\prime }}^{2}\right )^{3}=0} \]
✓ Solution by Maple
Time used: 0.14 (sec). Leaf size: 131
dsolve((y(x)*diff(y(x),x$2)+1+diff(y(x),x)^2)^2=(1+diff(y(x),x)^2)^3,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 ) = 0 \\ y \left (x \right ) = -c_{1} -\sqrt {c_{1}^{2}-c_{2}^{2}-2 x c_{2} -x^{2}} \\ y \left (x \right ) = -c_{1} +\sqrt {c_{1}^{2}-c_{2}^{2}-2 x c_{2} -x^{2}} \\ y \left (x \right ) = c_{1} -\sqrt {c_{1}^{2}-c_{2}^{2}-2 x c_{2} -x^{2}} \\ y \left (x \right ) = c_{1} +\sqrt {c_{1}^{2}-c_{2}^{2}-2 x c_{2} -x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 6.299 (sec). Leaf size: 121
DSolve[(y[x]*y''[x]+1+(y'[x])^2)^2==(1+(y'[x])^2)^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\sqrt {e^{2 c_1}-(x+c_2){}^2}-e^{c_1} \\ y(x)\to e^{c_1}-\sqrt {e^{2 c_1}-(x+c_2){}^2} \\ y(x)\to \sqrt {e^{2 c_1}-(x+c_2){}^2}-e^{c_1} \\ y(x)\to \sqrt {e^{2 c_1}-(x+c_2){}^2}+e^{c_1} \\ \end{align*}