Internal problem ID [2003]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 10, page 41
Problem number: 17.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_linear]
\[ \boxed {\sin \left (\theta \right ) r^{\prime }+\tan \left (\theta \right ) r=\cos \left (\theta \right )-1} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 24
dsolve(sin(theta)*diff(r(theta),theta)+1+r(theta)*tan(theta)=cos(theta),r(theta), singsol=all)
\[ r \left (\theta \right ) = \frac {2 \ln \left (\tan \left (\frac {\theta }{2}\right )-1\right )+\theta +c_{1}}{\sec \left (\theta \right )+\tan \left (\theta \right )} \]
✓ Solution by Mathematica
Time used: 6.359 (sec). Leaf size: 171
DSolve[Sin[\[Theta]]*r'[\[Theta]]+1+r[\[Theta]]*Tan[\[Theta]]==Cos[\[Theta]],r[\[Theta]],\[Theta],IncludeSingularSolutions -> True]
\[ r(\theta )\to \frac {1}{4} e^{-\coth ^{-1}(\sin (\theta ))} \left (-\frac {\sqrt {2} \sqrt {-\cot ^2(\theta )} \left (\frac {2 \left (\sqrt {\sin ^2(\theta )}-1\right ) \left (\sqrt {\cos (2 \theta )-1} \text {arctanh}\left (\sqrt {\cos ^2(\theta )}\right )+\sqrt {2} \sqrt {\sin ^2(\theta )} \log \left (\sqrt {\cos (2 \theta )+1}-\sqrt {\cos (2 \theta )-1}\right )\right )}{\sqrt {-\sin ^2(\theta )} (\csc (\theta )-1)}+2 \sqrt {\cos (2 \theta )+1} \tan (\theta ) \left (2 \log \left (1-\tan \left (\frac {\theta }{2}\right )\right )-\log \left (\tan \left (\frac {\theta }{2}\right )\right )\right )\right )}{\sqrt {\cos ^2(\theta )}}+4 c_1\right ) \]