48.4.10 problem Problem 3.14

Internal problem ID [7552]
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 3.14
Date solved : Monday, January 27, 2025 at 03:05:37 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} \left (\phi ^{\prime }-\frac {\phi ^{2}}{2}\right ) \sin \left (\theta \right )^{2}-\phi \sin \left (\theta \right ) \cos \left (\theta \right )&=\frac {\cos \left (2 \theta \right )}{2}+1 \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 37

dsolve((diff(phi(theta),theta)-1/2*phi(theta)^2)*sin(theta)^2-phi(theta)*sin(theta)*cos(theta)=1/2*cos(2*theta)+1,phi(theta), singsol=all)
 

\[ \phi = \frac {-\sinh \left (\frac {\theta }{2}\right ) c_{1} -\cosh \left (\frac {\theta }{2}\right )}{\cosh \left (\frac {\theta }{2}\right ) c_{1} +\sinh \left (\frac {\theta }{2}\right )}-\cot \left (\theta \right ) \]

✓ Solution by Mathematica

Time used: 0.538 (sec). Leaf size: 36

DSolve[(D[ \[Phi][\[Theta]] ,  \[Theta] ]-1/2\[Phi][\[Theta]]^2)*Sin[\[Theta]]^2-\[Phi][\[Theta]]*Sin[\[Theta]]*Cos[\[Theta]]==1/2*Cos[2*\[Theta]]+1,\[Phi][\[Theta]],\[Theta],IncludeSingularSolutions -> True]
 

\begin{align*} \phi (\theta )\to -\cot (\theta )-\frac {2 e^{\theta }}{e^{\theta }-2 c_1}+1 \\ \phi (\theta )\to 1-\cot (\theta ) \\ \end{align*}