89.10.12 problem 12

Internal problem ID [24505]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 4. Additional topics on equations of first order and first degree. Exercises at page 77
Problem number : 12
Date solved : Thursday, October 02, 2025 at 10:43:47 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

\begin{align*} \left (2 x -1\right ) y+2 \left (x^{2}+y^{2}-x \right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 125

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

\begin{align*} y &= \sqrt {-x^{2}+x -\sqrt {x^{4}-2 x^{3}+x^{2}-2 c_1}} \\ y &= \sqrt {-x^{2}+x +\sqrt {x^{4}-2 x^{3}+x^{2}-2 c_1}} \\ y &= -\sqrt {-x^{2}+x -\sqrt {x^{4}-2 x^{3}+x^{2}-2 c_1}} \\ y &= -\sqrt {-x^{2}+x +\sqrt {x^{4}-2 x^{3}+x^{2}-2 c_1}} \\ \end{align*}

✓ Mathematica. Time used: 2.314 (sec). Leaf size: 154

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

\begin{align*} y(x)&\to -\sqrt {-x^2-\sqrt {x^4-2 x^3+x^2+2 c_1}+x}\\ y(x)&\to \sqrt {-x^2-\sqrt {x^4-2 x^3+x^2+2 c_1}+x}\\ y(x)&\to -\sqrt {-x^2+\sqrt {x^4-2 x^3+x^2+2 c_1}+x}\\ y(x)&\to \sqrt {-x^2+\sqrt {x^4-2 x^3+x^2+2 c_1}+x}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 2.337 (sec). Leaf size: 112

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

\[ \left [ y{\left (x \right )} = - \sqrt {- x^{2} + x - \sqrt {C_{1} + x^{4} - 2 x^{3} + x^{2}}}, \ y{\left (x \right )} = \sqrt {- x^{2} + x - \sqrt {C_{1} + x^{4} - 2 x^{3} + x^{2}}}, \ y{\left (x \right )} = - \sqrt {- x^{2} + x + \sqrt {C_{1} + x^{4} - 2 x^{3} + x^{2}}}, \ y{\left (x \right )} = \sqrt {- x^{2} + x + \sqrt {C_{1} + x^{4} - 2 x^{3} + x^{2}}}\right ] \]