50.8.18 problem 4(b)

Internal problem ID [7934]
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)
Date solved : Monday, January 27, 2025 at 03:33:14 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} x y^{\prime \prime }&=y^{\prime }-2 {y^{\prime }}^{3} \end{align*}

✓ Solution by Maple

Time used: 0.039 (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 &= \frac {\sqrt {2 x^{2}-c_{1}}}{2}+c_{2} \\ y &= -\frac {\sqrt {2 x^{2}-c_{1}}}{2}+c_{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.735 (sec). Leaf size: 96

DSolve[x*D[y[x],{x,2}]==D[y[x],x]-2*(D[y[x],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*}