49.23.9 problem 5(b)

Internal problem ID [7767]
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)
Date solved : Sunday, March 30, 2025 at 12:23:06 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*}

With initial conditions

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

✓ Maple. Time used: 0.885 (sec). Leaf size: 49

ode:=diff(diff(y(x),x),x)+sin(y(x)) = 0; 
ic:=y(0) = 0, D(y)(0) = beta; 
dsolve([ode,ic],y(x), singsol=all);
 

\begin{align*} y &= \operatorname {RootOf}\left (-\int _{0}^{\textit {\_Z}}\frac {1}{\sqrt {2 \cos \left (\textit {\_a} \right )+\beta ^{2}-2}}d \textit {\_a} +x \right ) \\ y &= \operatorname {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*}

✓ Mathematica. Time used: 0.329 (sec). Leaf size: 19

ode=D[y[x],{x,2}]+Sin[y[x]]==0; 
ic={y[0]==0,Derivative[1][y][0] ==\[Beta]}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to 2 \operatorname {JacobiAmplitude}\left (\frac {x \beta }{2},\frac {4}{\beta ^2}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(sin(y(x)) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): beta} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : property object is not iterable