Internal problem ID [5650]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 2. Problems for Review and Discovery. Problems for Discussion and Exploration.
Page 105
Problem number: 4.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\sin \relax (y)=0} \end {gather*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 47
dsolve(diff(y(x),x$2)+sin(y(x))=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}\frac {1}{\sqrt {2 \cos \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {1}{\sqrt {2 \cos \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.068 (sec). Leaf size: 69
DSolve[y''[x]+Sin[y[x]]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -2 \text {am}\left (\frac {1}{2} \sqrt {(c_1+2) (x+c_2){}^2}|\frac {4}{c_1+2}\right ) \\ y(x)\to 2 \text {am}\left (\frac {1}{2} \sqrt {(c_1+2) (x+c_2){}^2}|\frac {4}{c_1+2}\right ) \\ \end{align*}