38.2.47 problem 47

Internal problem ID [6476]
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 : Monday, January 27, 2025 at 02:07:15 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*}

✓ Solution by Maple

Time used: 0.094 (sec). Leaf size: 39

dsolve([r(theta)*tan(theta)/(a^2-r(theta)^2)*diff(r(theta),theta)=1,r(1/4*Pi) = 0],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*}

✓ Solution by Mathematica

Time used: 0.148 (sec). Leaf size: 51

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