89.2.28 problem 30

Internal problem ID [24293]
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 27
Problem number : 30
Date solved : Thursday, October 02, 2025 at 10:08:08 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class C`], _dAlembert]

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

With initial conditions

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 25

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

\[ y = -\frac {\left (-1+\sqrt {9-4 \ln \left (x \right )}\right ) x}{2 \ln \left (x \right )-4} \]

✓ Mathematica. Time used: 0.22 (sec). Leaf size: 30

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

\begin{align*} y(x)&\to \frac {x-x \sqrt {9-4 \log (x)}}{2 (\log (x)-2)} \end{align*}

✗ Sympy

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

TypeError : cannot determine truth value of Relational: _X0**2 < 2