57.4.6 problem 1(f)

Internal problem ID [14330]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.3.1 Separable equations. Exercises page 26
Problem number : 1(f)
Date solved : Thursday, October 02, 2025 at 09:31:39 AM
CAS classification : [_quadrature]

\begin{align*} Q^{\prime }&=\frac {Q}{4+Q^{2}} \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 38

ode:=diff(Q(t),t) = Q(t)/(4+Q(t)^2); 
dsolve(ode,Q(t), singsol=all);
 

\[ Q = \frac {2 \,{\mathrm e}^{\frac {c_1}{4}+\frac {t}{4}}}{\sqrt {\frac {{\mathrm e}^{\frac {c_1}{2}+\frac {t}{2}}}{\operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {c_1}{2}+\frac {t}{2}}}{4}\right )}}} \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 42

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

\begin{align*} Q(t)&\to -\frac {\sqrt {t+4 c_1}}{\sqrt {2}}\\ Q(t)&\to \frac {\sqrt {t+4 c_1}}{\sqrt {2}} \end{align*}

✓ Sympy. Time used: 0.708 (sec). Leaf size: 49

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

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