89.4.20 problem 21

Internal problem ID [24342]
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. Exercises at page 39
Problem number : 21
Date solved : Thursday, October 02, 2025 at 10:20:37 PM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

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

With initial conditions

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

✓ Maple. Time used: 0.121 (sec). Leaf size: 21

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

\[ y = \frac {x^{2}}{2}+\frac {\sqrt {x^{4}+8 x}}{2} \]

✓ Mathematica. Time used: 0.36 (sec). Leaf size: 35

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

\begin{align*} y(x)&\to \frac {1}{2} x \left (x-\sqrt {-\frac {1}{x^2}} \sqrt {-x \left (x^3+8\right )}\right ) \end{align*}

✓ Sympy. Time used: 5.427 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = \frac {x^{2} \left (\sqrt {1 + \frac {8}{x^{3}}} + 1\right )}{2} \]