Internal problem ID [4423]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson
2018.
Section: Chapter 2, First order differential equations. Section 2.2, Separable Equations. Exercises. page
46
Problem number: 21.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {\frac {y^{\prime }}{\theta }-\frac {y \sin \left (\theta \right )}{y^{2}+1}=0} \] With initial conditions \begin {align*} [y \left (\pi \right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.531 (sec). Leaf size: 35
dsolve([1/theta*diff(y(theta),theta)= y(theta)*sin(theta)/(y(theta)^2+1),y(Pi) = 1],y(theta), singsol=all)
\[ y \left (\theta \right ) = \frac {{\mathrm e}^{-\theta \cos \left (\theta \right )+\sin \left (\theta \right )+\frac {1}{2}}}{\sqrt {\frac {{\mathrm e}^{-2 \theta \cos \left (\theta \right )+2 \sin \left (\theta \right )+1}}{\operatorname {LambertW}\left ({\mathrm e}^{-2 \theta \cos \left (\theta \right )-2 \pi +2 \sin \left (\theta \right )+1}\right )}}} \]
✓ Solution by Mathematica
Time used: 3.725 (sec). Leaf size: 26
DSolve[{1/\[Theta]*y'[\[Theta]]== y[\[Theta]]*Sin[\[Theta]]/(y[\[Theta]]^2+1),{y[Pi]==1}},y[\[Theta]],\[Theta],IncludeSingularSolutions -> True]
\begin{align*} y(\theta )\to \sqrt {W\left (e^{2 \sin (\theta )-2 \theta \cos (\theta )-2 \pi +1}\right )} \\ \end{align*}