Internal problem ID [5131]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]], _Riccati]
Solve \begin {gather*} \boxed {\left (\phi ^{\prime }-\frac {\phi ^{2}}{2}\right ) \left (\sin ^{2}\left (\theta \right )\right )-\phi \sin \left (\theta \right ) \cos \left (\theta \right )-\frac {\cos \left (2 \theta \right )}{2}-1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 39
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)
\[ \phi \left (\theta \right ) = -\frac {\sinh \left (\frac {\theta }{2}\right ) c_{1}+\cosh \left (\frac {\theta }{2}\right )}{\cosh \left (\frac {\theta }{2}\right ) c_{1}+\sinh \left (\frac {\theta }{2}\right )}-\frac {\cos \left (\theta \right )}{\sin \left (\theta \right )} \]
✓ Solution by Mathematica
Time used: 1.032 (sec). Leaf size: 35
DSolve[(\[Phi]'[\[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 {1}{-\frac {1}{2}+c_1 e^{-\theta }}+1 \\ \phi (\theta )\to 1-\cot (\theta ) \\ \end{align*}