31.2.9 problem 10.3.9 (b)

Internal problem ID [7694]
Book : Basic Training in Mathematics. By R. Shankar. Plenum Press. NY. 1995
Section : Chapter 10, Differential equations. Section 10.3, ODEs with variable Coefficients. First order. page 315
Problem number : 10.3.9 (b)
Date solved : Tuesday, September 30, 2025 at 04:56:08 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

\begin{align*} 3 x y^{\prime }+y+x^{2} y^{4}&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 88

ode:=3*x*diff(y(x),x)+y(x)+x^2*y(x)^4 = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.231 (sec). Leaf size: 61

ode=3*x*D[y[x],x]+y[x]+x^2*y[x]^4==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{\sqrt [3]{x (x+c_1)}}\\ y(x)&\to -\frac {\sqrt [3]{-1}}{\sqrt [3]{x (x+c_1)}}\\ y(x)&\to \frac {(-1)^{2/3}}{\sqrt [3]{x (x+c_1)}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 1.296 (sec). Leaf size: 58

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

\[ \left [ y{\left (x \right )} = \frac {\sqrt [3]{\frac {1}{x \left (C_{1} + x\right )}} \left (-1 - \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \frac {\sqrt [3]{\frac {1}{x \left (C_{1} + x\right )}} \left (-1 + \sqrt {3} i\right )}{2}, \ y{\left (x \right )} = \sqrt [3]{\frac {1}{x \left (C_{1} + x\right )}}\right ] \]