64.3.8 problem 9

Internal problem ID [13208]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.1 (Exact differential equations and integrating factors). Exercises page 37
Problem number : 9
Date solved : Monday, March 31, 2025 at 07:37:49 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 31

ode:=(2*s(t)-1)/t*diff(s(t),t)+(s(t)-s(t)^2)/t^2 = 0; 
dsolve(ode,s(t), singsol=all);
 

\begin{align*} s &= \frac {1}{2}-\frac {\sqrt {4 t c_1 +1}}{2} \\ s &= \frac {1}{2}+\frac {\sqrt {4 t c_1 +1}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.365 (sec). Leaf size: 47

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

\begin{align*} s(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {2 K[1]-1}{(K[1]-1) K[1]}dK[1]\&\right ][\log (t)+c_1] \\ s(t)\to 0 \\ s(t)\to 1 \\ \end{align*}

✓ Sympy. Time used: 0.569 (sec). Leaf size: 31

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

\[ \left [ s{\left (t \right )} = \frac {1}{2} - \frac {\sqrt {C_{1} t + 1}}{2}, \ s{\left (t \right )} = \frac {\sqrt {C_{1} t + 1}}{2} + \frac {1}{2}\right ] \]