Internal problem ID [12572]
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.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {x^{\prime \prime }-\left (2 \cos \left (x\right )-1\right ) \sin \left (x\right )=0} \]
✓ 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*}