30.6.19 problem 20

Internal problem ID [7554]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Review problems. page 79
Problem number : 20
Date solved : Tuesday, September 30, 2025 at 04:48:52 PM
CAS classification : [[_homogeneous, `class G`], _rational, _Bernoulli]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 74

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

\begin{align*} y &= \frac {\left (-300 x^{5}+125 c_1 \right )^{{1}/{3}}}{5 x} \\ y &= -\frac {\left (-300 x^{5}+125 c_1 \right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{10 x} \\ y &= \frac {\left (-300 x^{5}+125 c_1 \right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{10 x} \\ \end{align*}

✓ Mathematica. Time used: 0.147 (sec). Leaf size: 80

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

\begin{align*} y(x)&\to -\frac {\sqrt [3]{-\frac {1}{5}} \sqrt [3]{-12 x^5+5 c_1}}{x}\\ y(x)&\to \frac {\sqrt [3]{-\frac {12 x^5}{5}+c_1}}{x}\\ y(x)&\to \frac {(-1)^{2/3} \sqrt [3]{-\frac {12 x^5}{5}+c_1}}{x} \end{align*}

✓ Sympy. Time used: 0.610 (sec). Leaf size: 85

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

\[ \left [ y{\left (x \right )} = \frac {5^{\frac {2}{3}} \sqrt [3]{\frac {C_{1}}{x^{3}} - 12 x^{2}}}{5}, \ y{\left (x \right )} = \frac {5^{\frac {2}{3}} \left (-1 - \sqrt {3} i\right ) \sqrt [3]{\frac {C_{1}}{x^{3}} - 12 x^{2}}}{10}, \ y{\left (x \right )} = \frac {5^{\frac {2}{3}} \left (-1 + \sqrt {3} i\right ) \sqrt [3]{\frac {C_{1}}{x^{3}} - 12 x^{2}}}{10}\right ] \]