15.6.17 problem 17

Internal problem ID [2974]
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
Date solved : Monday, January 27, 2025 at 07:05:05 AM
CAS classification : [_linear]

\begin{align*} \sin \left (\theta \right ) r^{\prime }+1+r \tan \left (\theta \right )&=\cos \left (\theta \right ) \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 24

dsolve(sin(theta)*diff(r(theta),theta)+1+r(theta)*tan(theta)=cos(theta),r(theta), singsol=all)
 

\[ r = \frac {2 \ln \left (\tan \left (\frac {\theta }{2}\right )-1\right )+\theta +c_{1}}{\sec \left (\theta \right )+\tan \left (\theta \right )} \]

✓ Solution by Mathematica

Time used: 7.795 (sec). Leaf size: 164

DSolve[Sin[\[Theta]]*D[ r[\[Theta]], \[Theta] ]+1+r[\[Theta]]*Tan[\[Theta]]==Cos[\[Theta]],r[\[Theta]],\[Theta],IncludeSingularSolutions -> True]
 

\[ r(\theta )\to \frac {1}{4} e^{-\coth ^{-1}(\sin (\theta ))} \left (\frac {\sqrt {2} \sqrt {-\cot ^2(\theta )} \left (\frac {4 \sqrt {-\sin ^2(\theta )} \left (\sqrt {\sin ^2(\theta )}-1\right ) \cos (\theta ) \csc (2 \theta ) \left (\sqrt {2} \sqrt {\sin ^2(\theta )} \text {arctanh}\left (\sqrt {-\tan ^2(\theta )}\right )-\sqrt {\cos (2 \theta )-1} \text {arctanh}\left (\sqrt {\cos ^2(\theta )}\right )\right )}{\sin (\theta )-1}-2 \sqrt {\cos (2 \theta )+1} \tan (\theta ) \left (2 \log \left (1-\tan \left (\frac {\theta }{2}\right )\right )-\log \left (\tan \left (\frac {\theta }{2}\right )\right )\right )\right )}{\sqrt {\cos ^2(\theta )}}+4 c_1\right ) \]