15.22.13 problem 13

Internal problem ID [3347]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 40, page 186
Problem number : 13
Date solved : Sunday, March 30, 2025 at 01:37:34 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=\frac {\pi }{4}\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 26

Order:=7; 
ode:=diff(diff(y(x),x),x) = sin(y(x)); 
ic:=y(0) = 1/4*Pi, D(y)(0) = 0; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = \frac {\pi }{4}+\frac {1}{4} \sqrt {2}\, x^{2}+\frac {1}{48} x^{4}-\frac {1}{1440} \sqrt {2}\, x^{6}+\operatorname {O}\left (x^{7}\right ) \]

✓ Mathematica. Time used: 0.156 (sec). Leaf size: 40

ode=D[y[x],{x,2}]==Sin[y[x]]; 
ic={y[0]==Pi/4,Derivative[1][y][0] ==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,6}]
 

\[ y(x)\to -\frac {x^6}{720 \sqrt {2}}+\frac {x^4}{48}+\frac {x^2}{2 \sqrt {2}}+\frac {\pi }{4} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sin(y(x)) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): pi/4, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=7)
 

ValueError : ODE -sin(y(x)) + Derivative(y(x), (x, 2)) does not match hint 2nd_power_series_regular