8.18 problem 4(b)

Internal problem ID [5513]

Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Problems for Review and Discovery. Page 53
Problem number: 4(b).
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 {x y^{\prime \prime }-y^{\prime }+2 \left (y^{\prime }\right )^{3}=0} \end {gather*}

Solution by Maple

Time used: 0.11 (sec). Leaf size: 37

dsolve(x*diff(y(x),x$2)=diff(y(x),x)-2*(diff(y(x),x))^3,y(x), singsol=all)
 

\begin{align*} y \relax (x ) = \frac {\sqrt {2 x^{2}-c_{1}}}{2}+c_{2} \\ y \relax (x ) = -\frac {\sqrt {2 x^{2}-c_{1}}}{2}+c_{2} \\ \end{align*}

Solution by Mathematica

Time used: 1.0 (sec). Leaf size: 96

DSolve[x*y''[x]==y'[x]-2*(y'[x])^3,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_2-\frac {1}{2} \sqrt {2 x^2+e^{2 c_1}} \\ y(x)\to \frac {1}{2} \sqrt {2 x^2+e^{2 c_1}}+c_2 \\ y(x)\to -\frac {\sqrt {x^2}}{\sqrt {2}}+c_2 \\ y(x)\to \frac {\sqrt {x^2}}{\sqrt {2}}+c_2 \\ \end{align*}