Internal problem ID [6078]
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: 12.
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 y^{\prime \prime }+\left (y^{\prime }\right )^{3}-\left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.118 (sec). Leaf size: 44
dsolve(y(x)*diff(y(x),x$2)+diff(y(x),x)^3-diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = 0 \\ y \relax (x ) = c_{1} \\ y \relax (x ) = {\mathrm e}^{-\frac {c_{1} \LambertW \left (\frac {{\mathrm e}^{\frac {c_{2}}{c_{1}}} {\mathrm e}^{\frac {x}{c_{1}}}}{c_{1}}\right )-c_{2}-x}{c_{1}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.114 (sec). Leaf size: 24
DSolve[y[x]*y''[x]+(y'[x])^3-(y'[x])^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\text {ProductLog}\left (c_1 e^{c_1 (x+\log (c_2))}\right )}{c_1} \\ \end{align*}