Internal problem ID [12924]
Internal file name [OUTPUT/11577_Tuesday_November_07_2023_11_27_17_PM_45474179/index.tex]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Exercises section 1.3 page 47
Problem number: 19 a(iii).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {\theta ^{\prime }+\frac {9 \cos \left (\theta \right )}{10}={\frac {11}{10}}} \]
Integrating both sides gives \begin {align*} \int \frac {1}{\frac {11}{10}-\frac {9 \cos \left (\theta \right )}{10}}d \theta &= t +c_{1}\\ \sqrt {10}\, \arctan \left (\tan \left (\frac {\theta }{2}\right ) \sqrt {10}\right )&=t +c_{1} \end {align*}
Solving for \(\theta \) gives these solutions \begin {align*} \theta _1&=2 \arctan \left (\frac {\tan \left (\frac {\left (t +c_{1} \right ) \sqrt {10}}{10}\right ) \sqrt {10}}{10}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \theta &= 2 \arctan \left (\frac {\tan \left (\frac {\left (t +c_{1} \right ) \sqrt {10}}{10}\right ) \sqrt {10}}{10}\right ) \\ \end{align*}
Verification of solutions
\[ \theta = 2 \arctan \left (\frac {\tan \left (\frac {\left (t +c_{1} \right ) \sqrt {10}}{10}\right ) \sqrt {10}}{10}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \theta ^{\prime }+\frac {9 \cos \left (\theta \right )}{10}=\frac {11}{10} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \theta ^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \theta ^{\prime }=\frac {11}{10}-\frac {9 \cos \left (\theta \right )}{10} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\theta ^{\prime }}{\frac {11}{10}-\frac {9 \cos \left (\theta \right )}{10}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {\theta ^{\prime }}{\frac {11}{10}-\frac {9 \cos \left (\theta \right )}{10}}d t =\int 1d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \sqrt {10}\, \arctan \left (\tan \left (\frac {\theta }{2}\right ) \sqrt {10}\right )=t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} \theta \\ {} & {} & \theta =2 \arctan \left (\frac {\tan \left (\frac {\left (t +c_{1} \right ) \sqrt {10}}{10}\right ) \sqrt {10}}{10}\right ) \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 21
dsolve(diff(theta(t),t)=1-cos(theta(t))+(1+cos(theta(t)))*(1/10),theta(t), singsol=all)
\[ \theta \left (t \right ) = 2 \arctan \left (\frac {\tan \left (\frac {\left (t +c_{1} \right ) \sqrt {10}}{10}\right ) \sqrt {10}}{10}\right ) \]
✓ Solution by Mathematica
Time used: 10.277 (sec). Leaf size: 55
DSolve[theta'[t]==1-Cos[theta[t]]+(1+Cos[theta[t]])*(1/10),theta[t],t,IncludeSingularSolutions -> True]
\begin{align*} \theta (t)\to 2 \arctan \left (\frac {\tan \left (\frac {t-10 c_1}{\sqrt {10}}\right )}{\sqrt {10}}\right ) \\ \theta (t)\to -\arccos \left (\frac {11}{9}\right ) \\ \theta (t)\to \arccos \left (\frac {11}{9}\right ) \\ \theta (t)\to \text {Interval}[\{-\pi ,\pi \}] \\ \end{align*}