67.7.15 problem 15

Internal problem ID [16460]
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 : 15
Date solved : Thursday, October 02, 2025 at 01:34:07 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 39

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

\[ y = \frac {2 \,{\mathrm e}^{-\frac {c_1}{4}-\frac {x}{4}}}{\sqrt {-\frac {{\mathrm e}^{-\frac {c_1}{2}-\frac {x}{2}}}{\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-\frac {c_1}{2}-\frac {x}{2}}}{4}\right )}}} \]

✓ Mathematica. Time used: 20.119 (sec). Leaf size: 246

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

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

✓ Sympy. Time used: 0.979 (sec). Leaf size: 56

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

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