64.4.3 problem 3

Internal problem ID [13221]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.2 (Separable equations). Exercises page 47
Problem number : 3
Date solved : Monday, March 31, 2025 at 07:39:52 AM
CAS classification : [_separable]

\begin{align*} 2 r \left (s^{2}+1\right )+\left (r^{4}+1\right ) s^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 15

ode:=2*r*(s(r)^2+1)+(r^4+1)*diff(s(r),r) = 0; 
dsolve(ode,s(r), singsol=all);
 

\[ s = -\tan \left (\arctan \left (r^{2}\right )+2 c_1 \right ) \]

✓ Mathematica. Time used: 0.373 (sec). Leaf size: 59

ode=2*r*(s[r]^2+1)+(r^4+1)*D[ s[r],r]==0; 
ic={}; 
DSolve[{ode,ic},s[r],r,IncludeSingularSolutions->True]
 

\begin{align*} s(r)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+1}dK[1]\&\right ]\left [\int _1^r-\frac {2 K[2]}{K[2]^4+1}dK[2]+c_1\right ] \\ s(r)\to -i \\ s(r)\to i \\ \end{align*}

✓ Sympy. Time used: 0.347 (sec). Leaf size: 10

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

\[ s{\left (r \right )} = \tan {\left (C_{1} - \operatorname {atan}{\left (r^{2} \right )} \right )} \]