Internal problem ID [5346]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 6. Existence and uniqueness of solutions to systems and nth order equations. Page
238
Problem number: 5(b).
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*} With initial conditions \begin {align*} [y \relax (0) = 0, y^{\prime }\relax (0) = \beta ] \end {align*}
✓ Solution by Maple
Time used: 0.806 (sec). Leaf size: 53
dsolve([diff(y(x),x$2)+sin(y(x))=0,y(0) = 0, D(y)(0) = beta],y(x), singsol=all)
\begin{align*} y \relax (x ) = \RootOf \left (-\left (\int _{0}^{\textit {\_Z}}\frac {1}{\sqrt {2 \cos \left (\textit {\_a} \right )+\beta ^{2}-2}}d \textit {\_a} \right )+x \right ) \\ y \relax (x ) = \RootOf \left (\int _{0}^{\textit {\_Z}}\frac {1}{\sqrt {2 \cos \left (\textit {\_a} \right )+\beta ^{2}-2}}d \textit {\_a} +x \right ) \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.121 (sec). Leaf size: 19
DSolve[{y''[x]+Sin[y[x]]==0,{y[0]==0,y'[0]==\[Beta]}},y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to 2 \text {am}\left (\frac {x \beta }{2}|\frac {4}{\beta ^2}\right ) \\ \end{align*}