85.33.15 problem 15

Internal problem ID [22638]
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 65
Problem number : 15
Date solved : Thursday, October 02, 2025 at 08:57:12 PM
CAS classification : [_exact, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class B`]]

\begin{align*} r^{2} \sin \left (t \right )&=\left (2 r \cos \left (t \right )+10\right ) r^{\prime } \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 35

ode:=r(t)^2*sin(t) = (2*r(t)*cos(t)+10)*diff(r(t),t); 
dsolve(ode,r(t), singsol=all);
 

\begin{align*} r &= \left (-5+\sqrt {\cos \left (t \right ) c_1 +25}\right ) \sec \left (t \right ) \\ r &= \left (-5-\sqrt {\cos \left (t \right ) c_1 +25}\right ) \sec \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.333 (sec). Leaf size: 47

ode=r[t]^2*Sin[t]==(2*r[t]*Cos[t]+10)*D[r[t],t]; 
ic={}; 
DSolve[{ode,ic},r[t],t,IncludeSingularSolutions->True]
 

\begin{align*} r(t)&\to -\left (\sec (t) \left (5+\sqrt {25+c_1 \cos (t)}\right )\right )\\ r(t)&\to \sec (t) \left (-5+\sqrt {25+c_1 \cos (t)}\right )\\ r(t)&\to 0 \end{align*}

✓ Sympy. Time used: 1.479 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
r = Function("r") 
ode = Eq(-(2*r(t)*cos(t) + 10)*Derivative(r(t), t) + r(t)**2*sin(t),0) 
ics = {} 
dsolve(ode,func=r(t),ics=ics)
 

\[ \left [ r{\left (t \right )} = \frac {\sqrt {C_{1} \cos {\left (t \right )} + 25} - 5}{\cos {\left (t \right )}}, \ r{\left (t \right )} = - \frac {\sqrt {C_{1} \cos {\left (t \right )} + 25} + 5}{\cos {\left (t \right )}}\right ] \]