81.3.4 problem 4-9

Internal problem ID [21503]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 4. Homogeneous differential equations.
Problem number : 4-9
Date solved : Thursday, October 02, 2025 at 07:42:31 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

\begin{align*} y^{\prime }&=\frac {2 y^{4}+x^{4}}{x y^{3}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 62

ode:=diff(y(x),x) = (2*y(x)^4+x^4)/x/y(x)^3; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \left (c_1 \,x^{4}-1\right )^{{1}/{4}} x \\ y &= -\left (c_1 \,x^{4}-1\right )^{{1}/{4}} x \\ y &= -i \left (c_1 \,x^{4}-1\right )^{{1}/{4}} x \\ y &= i \left (c_1 \,x^{4}-1\right )^{{1}/{4}} x \\ \end{align*}

✓ Mathematica. Time used: 0.599 (sec). Leaf size: 136

ode=D[y[x],x] ==(2*y[x]^4+x^4)/(x*y[x]^3); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.041 (sec). Leaf size: 68

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (x**4 + 2*y(x)**4)/(x*y(x)**3),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = - i \sqrt [4]{x^{4} \left (C_{1} x^{4} - 1\right )}, \ y{\left (x \right )} = i \sqrt [4]{x^{4} \left (C_{1} x^{4} - 1\right )}, \ y{\left (x \right )} = - \sqrt [4]{x^{4} \left (C_{1} x^{4} - 1\right )}, \ y{\left (x \right )} = \sqrt [4]{x^{4} \left (C_{1} x^{4} - 1\right )}\right ] \]