89.33.19 problem 20

Internal problem ID [25003]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 17. Special Equations of order Two. Exercises at page 251
Problem number : 20
Date solved : Thursday, October 02, 2025 at 11:46:13 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} 2 y^{\prime \prime }&=\sin \left (2 y\right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=\frac {\pi }{2} \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✗ Maple

ode:=2*diff(diff(y(x),x),x) = sin(2*y(x)); 
ic:=[y(0) = 1/2*Pi, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.301 (sec). Leaf size: 19

ode=D[y[x],{x,2}]==Sin[2*y[x]]; 
ic={y[0]==Pi/2,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\operatorname {JacobiAmplitude}(i x-\operatorname {EllipticK}(2),2) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sin(2*y(x)) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): pi/2, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out