38.2.23 problem 23

Internal problem ID [8241]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 1. Introduction to differential equations. Section 1.2 Initial value problems. Exercises 1.2 at page 19
Problem number : 23
Date solved : Tuesday, September 30, 2025 at 05:19:43 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 42

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

\[ y = {\mathrm e}^{\frac {2 \sqrt {3}\, \operatorname {RootOf}\left (-3 \tan \left (\textit {\_Z} \right ) x -2 \sqrt {3}\, {\mathrm e}^{\frac {2 \sqrt {3}\, \textit {\_Z}}{3}-c_1}+\sqrt {3}\, x \right )}{3}-c_1} \]

✓ Mathematica. Time used: 0.072 (sec). Leaf size: 42

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

\[ \text {Solve}\left [\frac {2 \arctan \left (\frac {\frac {2 y(x)}{x}-1}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (\frac {y(x)}{x}\right )=-\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.634 (sec). Leaf size: 31

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

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