44.6.1 problem 1(a)

Internal problem ID [9179]
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. Section 1.8. Integrating Factors. Page 32
Problem number : 1(a)
Date solved : Tuesday, September 30, 2025 at 06:12:14 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} \left (3 x^{2}-y^{2}\right ) y^{\prime }-2 x y&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 313

ode:=(3*x^2-y(x)^2)*diff(y(x),x)-2*x*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {1+\frac {\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}}{2}+\frac {2}{\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}}}{3 c_1} \\ y &= -\frac {\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{2}/{3}}-4 i \sqrt {3}-4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}+4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}} c_1} \\ y &= \frac {\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{2}/{3}}-4 i \sqrt {3}+4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}}-4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_1^{2}-4}\, c_1 -108 x^{2} c_1^{2}+8\right )^{{1}/{3}} c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.081 (sec). Leaf size: 46

ode=(3*x^2-y[x]^2)*D[y[x],x]-2*x*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {K[1]^2-3}{(K[1]-1) K[1] (K[1]+1)}dK[1]=-\log (x)+c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out