Internal problem ID [6072]
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: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
Solve \begin {gather*} \boxed {y^{2} y^{\prime \prime }+\left (y^{\prime }\right )^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 34
dsolve(y(x)^2*diff(y(x),x$2)+diff(y(x),x)^3=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = c_{1} \\ y \relax (x ) = {\mathrm e}^{-\LambertW \left (-c_{1} {\mathrm e}^{-c_{2}} {\mathrm e}^{-x}\right )-c_{2}-x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.139 (sec). Leaf size: 37
DSolve[y[x]^2*y''[x]+(y'[x])^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2 \left (1+\frac {1}{\text {InverseFunction}\left [-\frac {1}{\text {$\#$1}}-\log (\text {$\#$1})+\log (\text {$\#$1}+1)\&\right ][-x+c_1]}\right ) \\ \end{align*}