32.2.46 problem 47

Internal problem ID [7763]
Book : Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY. 2001
Section : Program 24. First order differential equations. Further problems 24. page 1068
Problem number : 47
Date solved : Tuesday, September 30, 2025 at 05:05:07 PM
CAS classification : [_separable]

\begin{align*} \frac {r \tan \left (\theta \right ) r^{\prime }}{a^{2}-r^{2}}&=1 \end{align*}

With initial conditions

\begin{align*} r \left (\frac {\pi }{4}\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.150 (sec). Leaf size: 39

ode:=r(theta)*tan(theta)/(a^2-r(theta)^2)*diff(r(theta),theta) = 1; 
ic:=[r(1/4*Pi) = 0]; 
dsolve([ode,op(ic)],r(theta), singsol=all);
 

\begin{align*} r &= -\frac {a \sqrt {2}\, \sqrt {-\cos \left (2 \theta \right )}\, \csc \left (\theta \right )}{2} \\ r &= \frac {a \sqrt {2}\, \sqrt {-\cos \left (2 \theta \right )}\, \csc \left (\theta \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.111 (sec). Leaf size: 51

ode=r[\[Theta]]*Tan[\[Theta]]/(a^2-r[\[Theta]]^2)*D[ r[\[Theta]], \[Theta] ]==1; 
ic={r[Pi/4]==0}; 
DSolve[{ode,ic},r[\[Theta]],\[Theta],IncludeSingularSolutions->True]
 

\begin{align*} r(\theta )&\to -\sqrt {\frac {a^2 \cos (2 \theta )}{\cos (2 \theta )-1}}\\ r(\theta )&\to \sqrt {\frac {a^2 \cos (2 \theta )}{\cos (2 \theta )-1}} \end{align*}

✓ Sympy. Time used: 0.673 (sec). Leaf size: 39

from sympy import * 
theta = symbols("theta") 
a = symbols("a") 
r = Function("r") 
ode = Eq(-1 + r(theta)*tan(theta)*Derivative(r(theta), theta)/(a**2 - r(theta)**2),0) 
ics = {r(pi/4): 0} 
dsolve(ode,func=r(theta),ics=ics)
 

\[ \left [ r{\left (\theta \right )} = - \sqrt {a^{2} - \frac {a^{2}}{2 \sin ^{2}{\left (\theta \right )}}}, \ r{\left (\theta \right )} = \sqrt {a^{2} - \frac {a^{2}}{2 \sin ^{2}{\left (\theta \right )}}}\right ] \]