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: 32.
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 (1+y^{2}\right ) y^{\prime \prime }+\left (y^{\prime }\right )^{3}+y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.209 (sec). Leaf size: 126
dsolve((1+y(x)^2)*diff(y(x),x$2)+diff(y(x),x)^3+diff(y(x),x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -i \\ y \relax (x ) = i \\ y \relax (x ) = c_{1} \\ y \relax (x ) = \frac {i c_{1}-i-{\mathrm e}^{-\frac {-c_{1}^{2}+2 c_{1}+c_{1}^{2} c_{2}+c_{1}^{2} x +4 \LambertW \left (-\frac {i {\mathrm e}^{-\frac {c_{2} c_{1}}{4}} {\mathrm e}^{-\frac {c_{1} x}{4}} {\mathrm e}^{\frac {c_{1}}{4}} {\mathrm e}^{-\frac {c_{2}}{2}} {\mathrm e}^{-\frac {x}{2}} {\mathrm e}^{-\frac {1}{2}} {\mathrm e}^{-\frac {c_{2}}{4 c_{1}}} {\mathrm e}^{-\frac {x}{4 c_{1}}} {\mathrm e}^{\frac {1}{4 c_{1}}} \left (c_{1}-1\right )}{4 c_{1}}\right ) c_{1}-1+2 c_{2} c_{1}+2 c_{1} x +c_{2}+x}{4 c_{1}}}}{c_{1}+1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.203 (sec). Leaf size: 42
DSolve[(1+y[x]^2)*y''[x]+(y'[x])^3+y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \csc (c_1) \sec (c_1) \text {ProductLog}\left (\sin (c_1) e^{-\left ((x+c_2) \cos ^2(c_1)\right )-\sin ^2(c_1)}\right )+\tan (c_1) \\ \end{align*}