Internal problem ID [13212]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 14. Higher order equations and the reduction of order method. Additional
exercises page 277
Problem number: 14.1 (g).
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]]
\[ \boxed {\left (y+1\right ) y^{\prime \prime }-{y^{\prime }}^{3}=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 31
dsolve((y(x)+1)*diff(y(x),x$2)=diff(y(x),x)^3,y(x), singsol=all)
\begin{align*} y \left (x \right ) = -1 y \left (x \right ) = c_{1} y \left (x \right ) = {\mathrm e}^{\operatorname {LambertW}\left (-\left (c_{1} +c_{2} +x \right ) {\mathrm e}^{-c_{1}} {\mathrm e}^{-1}\right )+c_{1} +1}-1 \end{align*}
✓ Solution by Mathematica
Time used: 0.438 (sec). Leaf size: 93
DSolve[(y[x]+1)*y''[x]==y'[x]^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}[\text {$\#$1}-(\text {$\#$1}+1) \log (\text {$\#$1}+1)+\text {$\#$1} (-c_1)\&][x+c_2] y(x)\to \text {InverseFunction}[\text {$\#$1}-(\text {$\#$1}+1) \log (\text {$\#$1}+1)+\text {$\#$1} (-(-c_1))\&][x+c_2] y(x)\to \text {InverseFunction}[\text {$\#$1}-(\text {$\#$1}+1) \log (\text {$\#$1}+1)+\text {$\#$1} (-c_1)\&][x+c_2] \end{align*}