Internal problem ID [6077]
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: 11.
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]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-2 y \left (y^{\prime }\right )^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 386
dsolve(diff(y(x),x$2)=2*y(x)*diff(y(x),x)^3,y(x), singsol=all)
\begin{align*} y \relax (x ) = c_{1} \\ y \relax (x ) = \frac {\left (-12 c_{2}-12 x +4 \sqrt {-4 c_{1}^{3}+9 x^{2}+18 c_{2} x +9 c_{2}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 c_{1}}{\left (-12 c_{2}-12 x +4 \sqrt {-4 c_{1}^{3}+9 x^{2}+18 c_{2} x +9 c_{2}^{2}}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (-12 c_{2}-12 x +4 \sqrt {-4 c_{1}^{3}+9 x^{2}+18 c_{2} x +9 c_{2}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {c_{1}}{\left (-12 c_{2}-12 x +4 \sqrt {-4 c_{1}^{3}+9 x^{2}+18 c_{2} x +9 c_{2}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-12 c_{2}-12 x +4 \sqrt {-4 c_{1}^{3}+9 x^{2}+18 c_{2} x +9 c_{2}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 c_{1}}{\left (-12 c_{2}-12 x +4 \sqrt {-4 c_{1}^{3}+9 x^{2}+18 c_{2} x +9 c_{2}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (-12 c_{2}-12 x +4 \sqrt {-4 c_{1}^{3}+9 x^{2}+18 c_{2} x +9 c_{2}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {c_{1}}{\left (-12 c_{2}-12 x +4 \sqrt {-4 c_{1}^{3}+9 x^{2}+18 c_{2} x +9 c_{2}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-12 c_{2}-12 x +4 \sqrt {-4 c_{1}^{3}+9 x^{2}+18 c_{2} x +9 c_{2}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 c_{1}}{\left (-12 c_{2}-12 x +4 \sqrt {-4 c_{1}^{3}+9 x^{2}+18 c_{2} x +9 c_{2}^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.468 (sec). Leaf size: 346
DSolve[y''[x]==2*y[x]*(y'[x])^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2}}-\frac {\sqrt [3]{\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2}}{\sqrt [3]{2}} \\ y(x)\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2\right ){}^{2/3}+\sqrt [3]{2} \left (-2-2 i \sqrt {3}\right ) c_1}{4 \sqrt [3]{\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2}} \\ y(x)\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2\right ){}^{2/3}+2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) c_1}{4 \sqrt [3]{\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2}} \\ \end{align*}