Internal problem ID [6068]
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: 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-x \left (y^{\prime }\right )^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.048 (sec). Leaf size: 37
dsolve(diff(y(x),x$2)=x*(diff(y(x),x))^3,y(x), singsol=all)
\begin{align*} y \relax (x ) = \arctan \left (\frac {x}{\sqrt {-x^{2}+c_{1}}}\right )+c_{2} \\ y \relax (x ) = -\arctan \left (\frac {x}{\sqrt {-x^{2}+c_{1}}}\right )+c_{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.871 (sec). Leaf size: 53
DSolve[y''[x]==x*(y'[x])^3,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-\text {ArcTan}\left (\frac {x}{\sqrt {-x^2+c_1}}\right ) \\ y(x)\to \text {ArcTan}\left (\frac {x}{\sqrt {-x^2+c_1}}\right )+c_2 \\ y(x)\to c_2 \\ \end{align*}