15.22.9 problem 9

Internal problem ID [3343]
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 : 9
Date solved : Tuesday, March 04, 2025 at 04:36:27 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\cos \left (x \right )+\sin \left (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 )&=\frac {\pi }{2} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 18

Order:=4; 
ode:=diff(y(x),x) = cos(x)+sin(y(x)); 
ic:=y(1/2*Pi) = 1/2*Pi; 
dsolve([ode,ic],y(x),type='series',x=1/2*Pi);
 

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 22

ode=D[y[x],x]==Cos[x]*Sin[y[x]]; 
ic={y[Pi/2]==Pi/2}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,Pi/2,3}]
 

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

✓ Sympy. Time used: 1.510 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sin(y(x)) - cos(x) + Derivative(y(x), x),0) 
ics = {y(pi/2): pi/2} 
dsolve(ode,func=y(x),ics=ics,hint="1st_power_series",x0=pi/2,n=4)
 

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