67.3.34 problem 4.7 (h)

Internal problem ID [16355]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 4. SEPARABLE FIRST ORDER EQUATIONS. Additional exercises. page 90
Problem number : 4.7 (h)
Date solved : Thursday, October 02, 2025 at 01:22:13 PM
CAS classification : [_separable]

\begin{align*} \left (-1+y^{2}\right ) y^{\prime }&=4 x y^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 59

ode:=(y(x)^2-1)*diff(y(x),x) = 4*x*y(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= x^{2}+2 c_1 -\sqrt {x^{4}+4 c_1 \,x^{2}+4 c_1^{2}-1} \\ y &= x^{2}+2 c_1 +\sqrt {x^{4}+4 c_1 \,x^{2}+4 c_1^{2}-1} \\ \end{align*}

✓ Mathematica. Time used: 0.263 (sec). Leaf size: 84

ode=(y[x]^2-1)*D[y[x],x]==4*x*y[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \left (2 x^2-\sqrt {4 x^4+4 c_1 x^2-4+c_1{}^2}+c_1\right )\\ y(x)&\to \frac {1}{2} \left (2 x^2+\sqrt {4 x^4+4 c_1 x^2-4+c_1{}^2}+c_1\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.363 (sec). Leaf size: 61

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x*y(x)**2 + (y(x)**2 - 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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