73.7.49 problem 49

Internal problem ID [15128]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 8. Review exercises for part of part II. page 143
Problem number : 49
Date solved : Thursday, March 13, 2025 at 05:47:15 AM
CAS classification : [_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 45

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

\begin{align*} y &= x^{2}-\sqrt {x^{4}-x^{2}-2 c_{1}} \\ y &= x^{2}+\sqrt {x^{4}-x^{2}-2 c_{1}} \\ \end{align*}

✓ Mathematica. Time used: 0.127 (sec). Leaf size: 66

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

\begin{align*} y(x)\to x^2-i \sqrt {-x^4+x^2-c_1} \\ y(x)\to x^2+i \sqrt {-x^4+x^2-c_1} \\ y(x)\to \frac {1}{2} \\ \end{align*}

✓ Sympy. Time used: 1.119 (sec). Leaf size: 34

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

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