70.1.18 problem 2.4 (v)

Internal problem ID [14308]
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)
Date solved : Tuesday, January 28, 2025 at 06:26:11 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} x^{\prime \prime }&=\left (2 \cos \left (x\right )-1\right ) \sin \left (x\right ) \end{align*}

✓ Solution by Maple

Time used: 0.042 (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 \\ -\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 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.037 (sec). Leaf size: 48

DSolve[D[x[t],{t,2}]==(2*Cos[x[t]]-1)*Sin[x[t]],x[t],t,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{x(t)}\frac {1}{\sqrt {c_1+2 \int _1^{K[2]}(\sin (2 K[1])-\sin (K[1]))dK[1]}}dK[2]{}^2=(t+c_2){}^2,x(t)\right ] \]