4.9 problem Problem 3.12

Internal problem ID [5883]
Internal file name [OUTPUT/5131_Sunday_June_05_2022_03_25_41_PM_45774792/index.tex]

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 3.12.
ODE order: 2.
ODE degree: 1.

The type(s) of ODE detected by this program : "second_order_ode_missing_y"

Maple gives the following as the ode type

[[_2nd_order, _missing_y]]

\[ \boxed {u^{\prime \prime }-\cot \left (\theta \right ) u^{\prime }=0} \]

4.9.1 Solving as second order ode missing y ode

This is second order ode with missing dependent variable \(u\). Let \begin {align*} p(\theta ) &= u^{\prime } \end {align*}

Then \begin {align*} p'(\theta ) &= u^{\prime \prime } \end {align*}

Hence the ode becomes \begin {align*} p^{\prime }\left (\theta \right )-\cot \left (\theta \right ) p \left (\theta \right ) = 0 \end {align*}

Which is now solve for \(p(\theta )\) as first order ode. In canonical form the ODE is \begin {align*} p' &= F(\theta ,p)\\ &= f( \theta ) g(p)\\ &= \cot \left (\theta \right ) p \end {align*}

Where \(f(\theta )=\cot \left (\theta \right )\) and \(g(p)=p\). Integrating both sides gives \begin {align*} \frac {1}{p} \,dp &= \cot \left (\theta \right ) \,d \theta \\ \int { \frac {1}{p} \,dp} &= \int {\cot \left (\theta \right ) \,d \theta }\\ \ln \left (p \right )&=\ln \left (\sin \left (\theta \right )\right )+c_{1}\\ p&={\mathrm e}^{\ln \left (\sin \left (\theta \right )\right )+c_{1}}\\ &=c_{1} \sin \left (\theta \right ) \end {align*}

Since \(p=u^{\prime }\) then the new first order ode to solve is \begin {align*} u^{\prime } = c_{1} \sin \left (\theta \right ) \end {align*}

Integrating both sides gives \begin {align*} u &= \int { c_{1} \sin \left (\theta \right )\,\mathop {\mathrm {d}\theta }}\\ &= -c_{1} \cos \left (\theta \right )+c_{2} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} u &= -c_{1} \cos \left (\theta \right )+c_{2} \\ \end{align*}

Verification of solutions

\[ u = -c_{1} \cos \left (\theta \right )+c_{2} \] Verified OK.

4.9.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & u^{\prime \prime }-\cot \left (\theta \right ) u^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & u^{\prime \prime } \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} v =u^{\prime }\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & v^{\prime }\left (\theta \right )-\cot \left (\theta \right ) v \left (\theta \right )=0 \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & v^{\prime }\left (\theta \right )=\cot \left (\theta \right ) v \left (\theta \right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {v^{\prime }\left (\theta \right )}{v \left (\theta \right )}=\cot \left (\theta \right ) \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} \theta \\ {} & {} & \int \frac {v^{\prime }\left (\theta \right )}{v \left (\theta \right )}d \theta =\int \cot \left (\theta \right )d \theta +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (v \left (\theta \right )\right )=\ln \left (\sin \left (\theta \right )\right )+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} v \left (\theta \right ) \\ {} & {} & v \left (\theta \right )={\mathrm e}^{c_{1}} \sin \left (\theta \right ) \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} v \left (\theta \right ) \\ {} & {} & v \left (\theta \right )={\mathrm e}^{c_{1}} \sin \left (\theta \right ) \\ \bullet & {} & \textrm {Make substitution}\hspace {3pt} v =u^{\prime } \\ {} & {} & u^{\prime }={\mathrm e}^{c_{1}} \sin \left (\theta \right ) \\ \bullet & {} & \textrm {Integrate both sides to solve for}\hspace {3pt} u \\ {} & {} & \int u^{\prime }d \theta =\int {\mathrm e}^{c_{1}} \sin \left (\theta \right )d \theta +c_{2} \\ \bullet & {} & \textrm {Compute integrals}\hspace {3pt} \\ {} & {} & u=-{\mathrm e}^{c_{1}} \cos \left (\theta \right )+c_{2} \end {array} \]

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
<- LODE missing y successful`
 

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 10

dsolve(diff(u(theta),theta$2)-cot(theta)*diff(u(theta),theta)=0,u(theta), singsol=all)
                                                                                    
                                                                                    
 

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

✓ Solution by Mathematica

Time used: 0.085 (sec). Leaf size: 13

DSolve[u''[\[Theta]]-Cot[\[Theta]]*u'[\[Theta]]==0,u[\[Theta]],\[Theta],IncludeSingularSolutions -> True]
 

\[ u(\theta )\to c_2 \cos (\theta )+c_1 \]