53.4.11 problem 12

Internal problem ID [8499]
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
Date solved : Monday, January 27, 2025 at 04:07:28 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} y y^{\prime \prime }+{y^{\prime }}^{3}-{y^{\prime }}^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.034 (sec). Leaf size: 36

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 &= 0 \\ y &= c_{1} \\ y &= {\mathrm e}^{\frac {-c_{1} \operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {x +c_{2}}{c_{1}}}}{c_{1}}\right )+c_{2} +x}{c_{1}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 21.093 (sec). Leaf size: 32

DSolve[y[x]*D[y[x],{x,2}]+(D[y[x],x])^3-(D[y[x],x])^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{c_1} W\left (e^{e^{-c_1} \left (x-e^{c_1} c_1+c_2\right )}\right ) \]