15.22.16 problem 16

Internal problem ID [3350]
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 : 16
Date solved : Sunday, March 30, 2025 at 01:37:39 AM
CAS classification : [NONE]

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

Using series method with expansion around

\begin{align*} \frac {\pi }{2} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 20

Order:=5; 
ode:=diff(diff(y(x),x),x) = cos(x*y(x)); 
ic:=y(1/2*Pi) = 1, D(y)(1/2*Pi) = 1; 
dsolve([ode,ic],y(x),type='series',x=1/2*Pi);
 

\[ y = 1+\left (x -\frac {\pi }{2}\right )+\left (-\frac {\pi }{12}-\frac {1}{6}\right ) \left (x -\frac {\pi }{2}\right )^{3}-\frac {1}{12} \left (x -\frac {\pi }{2}\right )^{4}+\operatorname {O}\left (\left (x -\frac {\pi }{2}\right )^{5}\right ) \]

✓ Mathematica. Time used: 0.147 (sec). Leaf size: 42

ode=D[y[x],{x,2}]==Cos[x*y[x]]; 
ic={y[Pi/2]==1,Derivative[1][y][Pi/2]==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,Pi/2,4}]
 

\[ y(x)\to -\frac {1}{12} \left (x-\frac {\pi }{2}\right )^4+\frac {1}{12} (-2-\pi ) \left (x-\frac {\pi }{2}\right )^3+x-\frac {\pi }{2}+1 \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-cos(x*y(x)) + Derivative(y(x), (x, 2)),0) 
ics = {y(pi/2): 1, Subs(Derivative(y(x), x), x, pi/2): 1} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=pi/2,n=5)
 

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