Internal problem ID [4625]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {\frac {r \tan \left (\theta \right ) r^{\prime }}{a^{2}-r^{2}}-1=0} \end {gather*} With initial conditions \begin {align*} \left [r \left (\frac {\pi }{4}\right ) = 0\right ] \end {align*}
✓ Solution by Maple
Time used: 0.115 (sec). Leaf size: 41
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 \left (\theta \right ) = -\frac {\sqrt {-4 \left (\cos ^{2}\left (\theta \right )\right )+2}\, a}{2 \sin \left (\theta \right )} \\ r \left (\theta \right ) = \frac {\sqrt {-4 \left (\cos ^{2}\left (\theta \right )\right )+2}\, a}{2 \sin \left (\theta \right )} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.15 (sec). Leaf size: 51
DSolve[{r[\[Theta]]*Tan[\[Theta]]/(a^2-r[\[Theta]]^2)*r'[\[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*}