50.16.3 problem 4

Internal problem ID [8071]
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
Date solved : Monday, January 27, 2025 at 03:43:06 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }+\sin \left (y\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 47

dsolve(diff(y(x),x$2)+sin(y(x))=0,y(x), singsol=all)
 

\begin{align*} \int _{}^{y}\frac {1}{\sqrt {2 \cos \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} -x -c_{2} &= 0 \\ -\int _{}^{y}\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: 3.971 (sec). Leaf size: 69

DSolve[D[y[x],{x,2}]+Sin[y[x]]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -2 \operatorname {JacobiAmplitude}\left (\frac {1}{2} \sqrt {(c_1+2) (x+c_2){}^2},\frac {4}{c_1+2}\right ) \\ y(x)\to 2 \operatorname {JacobiAmplitude}\left (\frac {1}{2} \sqrt {(c_1+2) (x+c_2){}^2},\frac {4}{c_1+2}\right ) \\ \end{align*}