89.4.5 problem 5

Internal problem ID [24327]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 39
Problem number : 5
Date solved : Thursday, October 02, 2025 at 10:18:13 PM
CAS classification : [[_homogeneous, `class G`], _rational]

\begin{align*} y \left (x^{4}-y^{2}\right )+x \left (x^{4}+y^{2}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 203

ode:=y(x)*(x^4-y(x)^2)+x*(x^4+y(x)^2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\left (\left (4 c_1 +4 \sqrt {4 x^{6}+c_1^{2}}\right )^{{2}/{3}}-4 x^{2}\right ) x}{2 \left (4 c_1 +4 \sqrt {4 x^{6}+c_1^{2}}\right )^{{1}/{3}}} \\ y &= -\frac {\left (1+i \sqrt {3}\right ) x \left (4 c_1 +4 \sqrt {4 x^{6}+c_1^{2}}\right )^{{1}/{3}}}{4}-\frac {\left (i \sqrt {3}-1\right ) x^{3}}{\left (4 c_1 +4 \sqrt {4 x^{6}+c_1^{2}}\right )^{{1}/{3}}} \\ y &= \frac {\left (i \sqrt {3}\, \left (4 c_1 +4 \sqrt {4 x^{6}+c_1^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}\, x^{2}-\left (4 c_1 +4 \sqrt {4 x^{6}+c_1^{2}}\right )^{{2}/{3}}+4 x^{2}\right ) x}{4 \left (4 c_1 +4 \sqrt {4 x^{6}+c_1^{2}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 46.376 (sec). Leaf size: 301

ode=y[x]*( x^4-y[x]^2)+x*(x^4+y[x]^2)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {-2 \sqrt [3]{2} x^4+\left (2 \sqrt {x^6 \left (4 x^6+9 c_1{}^2\right )}+6 c_1 x^3\right ){}^{2/3}}{2 \sqrt [3]{\sqrt {x^6 \left (4 x^6+9 c_1{}^2\right )}+3 c_1 x^3}}\\ y(x)&\to \frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x^4+i \left (\sqrt {3}+i\right ) \left (2 \sqrt {x^6 \left (4 x^6+9 c_1{}^2\right )}+6 c_1 x^3\right ){}^{2/3}}{4 \sqrt [3]{\sqrt {x^6 \left (4 x^6+9 c_1{}^2\right )}+3 c_1 x^3}}\\ y(x)&\to \frac {\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) x^4+\left (-1-i \sqrt {3}\right ) \left (2 \sqrt {x^6 \left (4 x^6+9 c_1{}^2\right )}+6 c_1 x^3\right ){}^{2/3}}{4 \sqrt [3]{\sqrt {x^6 \left (4 x^6+9 c_1{}^2\right )}+3 c_1 x^3}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x**4 + y(x)**2)*Derivative(y(x), x) + (x**4 - y(x)**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out