Internal problem ID [12571]
Internal file name [OUTPUT/11224_Wednesday_October_18_2023_10_01_22_PM_50942243/index.tex]
Book: Nonlinear Ordinary Differential Equations by D.W.Jordna and P.Smith. 4th edition 1999.
Oxford Univ. Press. NY
Section: Chapter 2. Plane autonomous systems and linearization. Problems page 79
Problem number: 2.4 (v).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_x", "second_order_ode_can_be_made_integrable"
Maple gives the following as the ode type
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {x^{\prime \prime }-\left (2 \cos \left (x\right )-1\right ) \sin \left (x\right )=0} \]
Multiplying the ode by \(x^{\prime }\) gives \[ x^{\prime } x^{\prime \prime }-x^{\prime } \left (\sin \left (2 x\right )-\sin \left (x\right )\right ) = 0 \] Integrating the above w.r.t \(t\) gives \begin {align*} \int \left (x^{\prime } x^{\prime \prime }-x^{\prime } \left (\sin \left (2 x\right )-\sin \left (x\right )\right )\right )d t &= 0 \\ \frac {{x^{\prime }}^{2}}{2}-\cos \left (x\right )+\frac {\cos \left (2 x\right )}{2} = c_2 \end {align*}
Which is now solved for \(x\). Solving the given ode for \(x^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} x^{\prime }&=\sqrt {1-2 \cos \left (x\right )^{2}+2 \cos \left (x\right )+2 c_{1}} \tag {1} \\ x^{\prime }&=-\sqrt {1-2 \cos \left (x\right )^{2}+2 \cos \left (x\right )+2 c_{1}} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} \int \frac {1}{\sqrt {1-2 \cos \left (x \right )^{2}+2 \cos \left (x \right )+2 c_{1}}}d x &= \int {dt}\\ \int _{}^{x}\frac {1}{\sqrt {1-2 \cos \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a}&= t +c_{2} \end {align*}
Solving equation (2)
Integrating both sides gives \begin {align*} \int -\frac {1}{\sqrt {1-2 \cos \left (x \right )^{2}+2 \cos \left (x \right )+2 c_{1}}}d x &= \int {dt}\\ \int _{}^{x}-\frac {1}{\sqrt {1-2 \cos \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a}&= t +c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{x}\frac {1}{\sqrt {1-2 \cos \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a} &= t +c_{2} \\ \tag{2} \int _{}^{x}-\frac {1}{\sqrt {1-2 \cos \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a} &= t +c_{3} \\ \end{align*}
Verification of solutions
\[ \int _{}^{x}\frac {1}{\sqrt {1-2 \cos \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a} = t +c_{2} \] Verified OK.
\[ \int _{}^{x}-\frac {1}{\sqrt {1-2 \cos \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a} = t +c_{3} \] Verified OK.
This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable \(x\) an independent variable. Using \begin {align*} x' &= p(x) \end {align*}
Then \begin {align*} x'' &= \frac {dp}{dt}\\ &= \frac {dx}{dt} \frac {dp}{dx}\\ &= p \frac {dp}{dx} \end {align*}
Hence the ode becomes \begin {align*} p \left (x \right ) \left (\frac {d}{d x}p \left (x \right )\right ) = \sin \left (2 x \right )-\sin \left (x \right ) \end {align*}
Which is now solved as first order ode for \(p(x)\). In canonical form the ODE is \begin {align*} p' &= F(x,p)\\ &= f( x) g(p)\\ &= \frac {\sin \left (2 x \right )-\sin \left (x \right )}{p} \end {align*}
Where \(f(x)=\sin \left (2 x \right )-\sin \left (x \right )\) and \(g(p)=\frac {1}{p}\). Integrating both sides gives \begin{align*} \frac {1}{\frac {1}{p}} \,dp &= \sin \left (2 x \right )-\sin \left (x \right ) \,d x \\ \int { \frac {1}{\frac {1}{p}} \,dp} &= \int {\sin \left (2 x \right )-\sin \left (x \right ) \,d x} \\ \frac {p^{2}}{2}&=\cos \left (x \right )-\frac {\cos \left (2 x \right )}{2}+c_{1} \\ \end{align*} The solution is \[ \frac {p \left (x \right )^{2}}{2}-\cos \left (x \right )+\frac {\cos \left (2 x \right )}{2}-c_{1} = 0 \] For solution (1) found earlier, since \(p=x^{\prime }\) then we now have a new first order ode to solve which is \begin {align*} \frac {{x^{\prime }}^{2}}{2}-\cos \left (x\right )+\frac {\cos \left (2 x\right )}{2}-c_{1} = 0 \end {align*}
Solving the given ode for \(x^{\prime }\) results in \(2\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} x^{\prime }&=\sqrt {1-2 \cos \left (x\right )^{2}+2 \cos \left (x\right )+2 c_{1}} \tag {1} \\ x^{\prime }&=-\sqrt {1-2 \cos \left (x\right )^{2}+2 \cos \left (x\right )+2 c_{1}} \tag {2} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} \int \frac {1}{\sqrt {1-2 \cos \left (x \right )^{2}+2 \cos \left (x \right )+2 c_{1}}}d x &= \int {dt}\\ \int _{}^{x}\frac {1}{\sqrt {1-2 \cos \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a}&= t +c_{2} \end {align*}
Solving equation (2)
Integrating both sides gives \begin {align*} \int -\frac {1}{\sqrt {1-2 \cos \left (x \right )^{2}+2 \cos \left (x \right )+2 c_{1}}}d x &= \int {dt}\\ \int _{}^{x}-\frac {1}{\sqrt {1-2 \cos \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a}&= t +c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{x}\frac {1}{\sqrt {1-2 \cos \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a} &= t +c_{2} \\ \tag{2} \int _{}^{x}-\frac {1}{\sqrt {1-2 \cos \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a} &= t +c_{3} \\ \end{align*}
Verification of solutions
\[ \int _{}^{x}\frac {1}{\sqrt {1-2 \cos \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a} = t +c_{2} \] Verified OK.
\[ \int _{}^{x}-\frac {1}{\sqrt {1-2 \cos \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+2 c_{1}}}d \textit {\_a} = t +c_{3} \] Verified OK.
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = exp_sym -> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)-sin(2*_a)+sin(_a) = 0, _b(_a)` *** Sublevel 2 *** Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli <- Bernoulli successful <- differential order: 2; canonical coordinates successful <- differential order 2; missing variables successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 59
dsolve(diff(x(t),t$2)=(2*cos(x(t))-1)*sin(x(t)),x(t), singsol=all)
\begin{align*} \int _{}^{x \left (t \right )}\frac {1}{\sqrt {2 \sin \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} -t -c_{2} &= 0 \\ -\left (\int _{}^{x \left (t \right )}\frac {1}{\sqrt {2 \sin \left (\textit {\_a} \right )^{2}+2 \cos \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} \right )-t -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 61.831 (sec). Leaf size: 437
DSolve[x''[t]==(2*Cos[x[t]]-1)*Sin[x[t]],x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -2 \arccos \left (-\frac {1}{2} \sqrt {3-\sqrt {3+2 c_1}}\right ) \\ x(t)\to 2 \arccos \left (-\frac {1}{2} \sqrt {3-\sqrt {3+2 c_1}}\right ) \\ x(t)\to -2 \arccos \left (\frac {1}{2} \sqrt {3-\sqrt {3+2 c_1}}\right ) \\ x(t)\to 2 \arccos \left (\frac {1}{2} \sqrt {3-\sqrt {3+2 c_1}}\right ) \\ x(t)\to -2 \arccos \left (-\frac {1}{2} \sqrt {3+\sqrt {3+2 c_1}}\right ) \\ x(t)\to 2 \arccos \left (-\frac {1}{2} \sqrt {3+\sqrt {3+2 c_1}}\right ) \\ x(t)\to -2 \arccos \left (\frac {1}{2} \sqrt {3+\sqrt {3+2 c_1}}\right ) \\ x(t)\to 2 \arccos \left (\frac {1}{2} \sqrt {3+\sqrt {3+2 c_1}}\right ) \\ x(t)\to -2 i \text {arctanh}\left (\frac {\text {sn}\left (\frac {1}{2} \sqrt {\left (-c_1+2 \sqrt {2 c_1+3}-3\right ) (t+c_2){}^2}|\frac {c_1+2 \sqrt {2 c_1+3}+3}{c_1-2 \sqrt {2 c_1+3}+3}\right )}{\sqrt {\frac {-3+c_1}{3+c_1+2 \sqrt {3+2 c_1}}}}\right ) \\ x(t)\to 2 i \text {arctanh}\left (\frac {\text {sn}\left (\frac {1}{2} \sqrt {\left (-c_1+2 \sqrt {2 c_1+3}-3\right ) (t+c_2){}^2}|\frac {c_1+2 \sqrt {2 c_1+3}+3}{c_1-2 \sqrt {2 c_1+3}+3}\right )}{\sqrt {\frac {-3+c_1}{3+c_1+2 \sqrt {3+2 c_1}}}}\right ) \\ \end{align*}