Internal problem ID [6859]
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]]
\[ \boxed {3 y y^{\prime } y^{\prime \prime }-{y^{\prime }}^{3}=-1} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 119
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 \left (x \right )+1\right )^{\frac {2}{3}}+\left (-2 x -2 c_{2} \right ) c_{1}}{2 c_{1}} &= 0 \\ \frac {-i \left (x +c_{2} \right ) c_{1} \sqrt {3}+\left (-x -c_{2} \right ) c_{1} -3 \left (c_{1} y \left (x \right )+1\right )^{\frac {2}{3}}}{c_{1} \left (1+i \sqrt {3}\right )} &= 0 \\ \frac {-3 i \left (c_{1} y \left (x \right )+1\right )^{\frac {2}{3}}+\left (-x -c_{2} \right ) c_{1} \sqrt {3}-i \left (x +c_{2} \right ) c_{1}}{c_{1} \left (\sqrt {3}+i\right )} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 45.036 (sec). Leaf size: 126
DSolve[3*y[x]*y'[x]*y''[x]==(y'[x])^3-1,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{9} e^{-3 c_1} \left (-9+2 \sqrt {6} \left (e^{3 c_1} (x+c_2)\right ){}^{3/2}\right ) \\ y(x)\to \frac {1}{9} e^{-3 c_1} \left (-9+2 \sqrt {6} \left (-\sqrt [3]{-1} e^{3 c_1} (x+c_2)\right ){}^{3/2}\right ) \\ y(x)\to \frac {1}{9} e^{-3 c_1} \left (-9+2 \sqrt {6} \left ((-1)^{2/3} e^{3 c_1} (x+c_2)\right ){}^{3/2}\right ) \\ \end{align*}