64.5.12 problem 12

Internal problem ID [13254]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 2, section 2.3 (Linear equations). Exercises page 56
Problem number : 12
Date solved : Monday, March 31, 2025 at 07:43:39 AM
CAS classification : [_linear]

\begin{align*} \cos \left (t \right ) r^{\prime }+r \sin \left (t \right )-\cos \left (t \right )^{4}&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 19

ode:=cos(t)*diff(r(t),t)+r(t)*sin(t)-cos(t)^4 = 0; 
dsolve(ode,r(t), singsol=all);
 

\[ r = \frac {\left (2 t +\sin \left (2 t \right )+4 c_1 \right ) \cos \left (t \right )}{4} \]

✓ Mathematica. Time used: 0.057 (sec). Leaf size: 23

ode=Cos[t]*D[r[t],t]+(r[t]*Sin[t]-Cos[t]^4)==0; 
ic={}; 
DSolve[{ode,ic},r[t],t,IncludeSingularSolutions->True]
 

\[ r(t)\to \cos (t) \left (\int _1^t\cos ^2(K[1])dK[1]+c_1\right ) \]

✓ Sympy. Time used: 0.469 (sec). Leaf size: 17

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

\[ r{\left (t \right )} = \left (C_{1} + \frac {t}{2} + \frac {\sin {\left (2 t \right )}}{4}\right ) \cos {\left (t \right )} \]