Internal problem ID [6106]
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: 42.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {3 y y^{\prime } y^{\prime \prime }-\left (y^{\prime }\right )^{3}+1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.17 (sec). Leaf size: 87
dsolve(3*y(x)*diff(y(x),x)*diff(y(x),x$2)=diff(y(x),x)^3-1,y(x), singsol=all)
\begin{align*} \frac {3 \left (c_{1} y \relax (x )+1\right )^{\frac {2}{3}}}{2 c_{1}}-x -c_{2} = 0 \\ \frac {3 \left (c_{1} y \relax (x )+1\right )^{\frac {2}{3}}}{c_{1} \left (-1+i \sqrt {3}\right )}-x -c_{2} = 0 \\ -\frac {3 \left (c_{1} y \relax (x )+1\right )^{\frac {2}{3}}}{c_{1} \left (1+i \sqrt {3}\right )}-x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.177 (sec). Leaf size: 114
DSolve[3*y[x]*y'[x]*y''[x]==(y'[x])^3-1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-9+2 \sqrt {6} \left (c_1{}^3 (x+c_2)\right ){}^{3/2}}{9 c_1{}^3} \\ y(x)\to \frac {-9+2 \sqrt {6} \left (-\sqrt [3]{-1} c_1{}^3 (x+c_2)\right ){}^{3/2}}{9 c_1{}^3} \\ y(x)\to \frac {-9+2 \sqrt {6} \left ((-1)^{2/3} c_1{}^3 (x+c_2)\right ){}^{3/2}}{9 c_1{}^3} \\ \end{align*}