Internal problem ID [9186]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1607.
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 }+a \sin \relax (y)=0} \end {gather*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 49
dsolve(diff(diff(y(x),x),x)+a*sin(y(x))=0,y(x), singsol=all)
\begin{align*} \int _{}^{y \relax (x )}\frac {1}{\sqrt {2 a \cos \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {1}{\sqrt {2 a \cos \left (\textit {\_a} \right )+c_{1}}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.066 (sec). Leaf size: 79
DSolve[a*Sin[y[x]] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -2 \text {am}\left (\frac {1}{2} \sqrt {(2 a+c_1) (x+c_2){}^2}|\frac {4 a}{2 a+c_1}\right ) \\ y(x)\to 2 \text {am}\left (\frac {1}{2} \sqrt {(2 a+c_1) (x+c_2){}^2}|\frac {4 a}{2 a+c_1}\right ) \\ \end{align*}