8.23 problem 23

Internal problem ID [4378]

Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section: Chapter 8, Ordinary differential equations. Section 13. Miscellaneous problems. page 466
Problem number: 23.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_linear]

Solve \begin {gather*} \boxed {\sin \left (\theta \right ) \cos \left (\theta \right ) r^{\prime }-\left (\sin ^{2}\left (\theta \right )\right )-r \left (\cos ^{2}\left (\theta \right )\right )=0} \end {gather*}

Solution by Maple

Time used: 0.016 (sec). Leaf size: 15

dsolve(sin(theta)*cos(theta)*diff(r(theta),theta)-sin(theta)^2=r(theta)*cos(theta)^2,r(theta), singsol=all)
 

\[ r \left (\theta \right ) = \left (\ln \left (\sec \left (\theta \right )+\tan \left (\theta \right )\right )+c_{1}\right ) \sin \left (\theta \right ) \]

Solution by Mathematica

Time used: 0.092 (sec). Leaf size: 43

DSolve[Sin[\[Theta]]*Cos[\[Theta]]*r'[\[Theta]]-Sin[\[Theta]]^2==r[\[Theta]]*Cos[\[Theta]]^2,r[\[Theta]],\[Theta],IncludeSingularSolutions -> True]
 

\begin{align*} r(\theta )\to \sin (\theta ) \left (-\log \left (\cos \left (\frac {\theta }{2}\right )-\sin \left (\frac {\theta }{2}\right )\right )+\log \left (\sin \left (\frac {\theta }{2}\right )+\cos \left (\frac {\theta }{2}\right )\right )+c_1\right ) \\ \end{align*}