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 : Wednesday, March 05, 2025 at 05:19:19 AM
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*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 37

ode:=x*diff(diff(y(x),x),x) = diff(y(x),x)-2*diff(y(x),x)^3; 
dsolve(ode,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*}

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

ode=x*D[y[x],{x,2}]==D[y[x],x]-2*(D[y[x],x])^3; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 10.959 (sec). Leaf size: 68

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + 2*Derivative(y(x), x)**3 - Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = C_{1} - \frac {C_{2} \sqrt {\frac {1}{C_{2} + 2 x^{2}}}}{2} - x^{2} \sqrt {\frac {1}{C_{2} + 2 x^{2}}}, \ y{\left (x \right )} = C_{1} + \frac {C_{2} \sqrt {\frac {1}{C_{2} + 2 x^{2}}}}{2} + x^{2} \sqrt {\frac {1}{C_{2} + 2 x^{2}}}\right ] \]