89.6.16 problem 16

Internal problem ID [24399]
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. Miscellaneous Exercises at page 45
Problem number : 16
Date solved : Thursday, October 02, 2025 at 10:24:45 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y \left (x^{2}+y^{2}\right )+x \left (3 x^{2}-5 y^{2}\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.746 (sec). Leaf size: 27

ode:=y(x)*(x^2+y(x)^2)+x*(3*x^2-5*y(x)^2)*diff(y(x),x) = 0; 
ic:=[y(2) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \operatorname {RootOf}\left (2 \textit {\_Z}^{20} x^{4}-2 \textit {\_Z}^{12} x^{4}+3\right )^{4} x \]

✓ Mathematica. Time used: 1.683 (sec). Leaf size: 27

ode=y[x]*( x^2+y[x]^2 )+ x*(3*x^2-5*y[x]^2 )*D[y[x],x]==0; 
ic={y[2]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {Root}\left [2 \text {$\#$1}^5-2 \text {$\#$1}^3 x^2+3 x\&,2\right ] \end{align*}

✓ Sympy. Time used: 0.466 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(3*x**2 - 5*y(x)**2)*Derivative(y(x), x) + (x**2 + y(x)**2)*y(x),0) 
ics = {y(2): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \log {\left (x \right )} = - \log {\left (\left (\frac {y{\left (x \right )}}{x}\right )^{\frac {3}{4}} \sqrt [4]{-1 + \frac {y^{2}{\left (x \right )}}{x^{2}}} \right )} - \frac {\log {\left (2 \right )}}{4} + \frac {\log {\left (3 \right )}}{4} + \frac {i \pi }{4} \]