73.9.7 problem 14.1 (g)

Internal problem ID [15268]
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)
Date solved : Tuesday, January 28, 2025 at 07:51:12 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} \left (y+1\right ) y^{\prime \prime }&={y^{\prime }}^{3} \end{align*}

✓ Solution by Maple

Time used: 0.095 (sec). Leaf size: 53

dsolve((y(x)+1)*diff(y(x),x$2)=diff(y(x),x)^3,y(x), singsol=all)
 

\begin{align*} y &= -1 \\ y &= c_{1} \\ y &= \frac {-c_{1} -c_{2} -x -\operatorname {LambertW}\left (-\left (c_{1} +c_{2} +x \right ) {\mathrm e}^{-c_{1} -1}\right )}{\operatorname {LambertW}\left (-\left (c_{1} +c_{2} +x \right ) {\mathrm e}^{-c_{1} -1}\right )} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.400 (sec). Leaf size: 93

DSolve[(y[x]+1)*D[y[x],{x,2}]==D[y[x],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*}