Internal problem ID [6097]
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: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {\left (y y^{\prime \prime }+1+\left (y^{\prime }\right )^{2}\right )^{2}-\left (1+\left (y^{\prime }\right )^{2}\right )^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.142 (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 \relax (x ) = -i x +c_{1} \\ y \relax (x ) = i x +c_{1} \\ y \relax (x ) = 0 \\ y \relax (x ) = -c_{1}-\sqrt {c_{1}^{2}-c_{2}^{2}-2 c_{2} x -x^{2}} \\ y \relax (x ) = -c_{1}+\sqrt {c_{1}^{2}-c_{2}^{2}-2 c_{2} x -x^{2}} \\ y \relax (x ) = c_{1}-\sqrt {c_{1}^{2}-c_{2}^{2}-2 c_{2} x -x^{2}} \\ y \relax (x ) = c_{1}+\sqrt {c_{1}^{2}-c_{2}^{2}-2 c_{2} x -x^{2}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 5.928 (sec). Leaf size: 113
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 {c_1-(x+c_2){}^2}-\sqrt {c_1} \\ y(x)\to \sqrt {c_1}-\sqrt {c_1-(x+c_2){}^2} \\ y(x)\to \sqrt {c_1-(x+c_2){}^2}-\sqrt {c_1} \\ y(x)\to \sqrt {c_1-(x+c_2){}^2}+\sqrt {c_1} \\ \end{align*}