85.15.6 problem 1 (f)

Internal problem ID [22531]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 47
Problem number : 1 (f)
Date solved : Thursday, October 02, 2025 at 08:46:47 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class B`]]

\begin{align*} r^{\prime }&=\frac {r \sin \left (t \right )}{2 r \cos \left (t \right )-1} \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 19

ode:=diff(r(t),t) = r(t)*sin(t)/(2*r(t)*cos(t)-1); 
dsolve(ode,r(t), singsol=all);
 

\[ {\mathrm e}^{2 r} \cos \left (t \right )+\operatorname {Ei}_{1}\left (-2 r\right )+c_1 = 0 \]

✓ Mathematica. Time used: 0.881 (sec). Leaf size: 23

ode=D[r[t],t]== (r[t]*Sin[t])/( 2*r[t]*Cos[t]-1); 
ic={}; 
DSolve[{ode,ic},r[t],t,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [e^{2 r(t)} \cos (t)-\operatorname {ExpIntegralEi}(2 r(t))=c_1,r(t)\right ] \]

✓ Sympy. Time used: 1.359 (sec). Leaf size: 19

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

\[ C_{1} - e^{2 r{\left (t \right )}} \cos {\left (t \right )} + \operatorname {Ei}{\left (2 r{\left (t \right )} \right )} = 0 \]