48.4.8 problem Problem 3.11

Internal problem ID [7550]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 3. Ordinary Differential Equations. Section 3.6 Summary and Problems. Page 218
Problem number : Problem 3.11
Date solved : Monday, January 27, 2025 at 03:05:33 PM
CAS classification : [_Bernoulli]

\begin{align*} \phi ^{\prime }-\frac {\phi ^{2}}{2}-\phi \cot \left (\theta \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 16

dsolve(diff(phi(theta),theta)-1/2*phi(theta)^2-phi(theta)*cot(theta)=0,phi(theta), singsol=all)
 

\[ \phi = \frac {2 \sin \left (\theta \right )}{\cos \left (\theta \right )+2 c_{1}} \]

✓ Solution by Mathematica

Time used: 0.279 (sec). Leaf size: 23

DSolve[D[ \[Phi][\[Theta]] ,  \[Theta] ]-1/2*\[Phi][\[Theta]]^2-\[Phi][\[Theta]]*Cot[\[Theta]]==0,\[Phi][\[Theta]],\[Theta],IncludeSingularSolutions -> True]
 

\begin{align*} \phi (\theta )\to \frac {2 \sin (\theta )}{\cos (\theta )+2 c_1} \\ \phi (\theta )\to 0 \\ \end{align*}