44.8.18 problem 4(b)

Internal problem ID [9220]
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 : Tuesday, September 30, 2025 at 06:15:19 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*}

✓ Maple. Time used: 0.025 (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: 0.471 (sec). Leaf size: 48

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 \int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (2 K[1]^2-1\right )}dK[1]\&\right ][c_1-\log (K[2])]dK[2]+c_2 \end{align*}

✓ Sympy. Time used: 6.872 (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)
 

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