52.1.34 problem 30 (b)

Internal problem ID [8251]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. Section 6.2 SOLUTIONS ABOUT ORDINARY POINTS. EXERCISES 6.2. Page 246
Problem number : 30 (b)
Date solved : Sunday, March 30, 2025 at 12:49:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 24

Order:=8; 
ode:=diff(diff(y(x),x),x)+y(x)*cos(x) = 0; 
ic:=y(0) = 1, D(y)(0) = 1; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = 1+x -\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}-\frac {1}{80} x^{6}-\frac {19}{5040} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 48

ode=D[y[x],{x,2}]+Cos[x]*y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to -\frac {19 x^7}{5040}-\frac {x^6}{80}+\frac {x^5}{30}+\frac {x^4}{12}-\frac {x^3}{6}-\frac {x^2}{2}+x+1 \]

✓ Sympy. Time used: 0.975 (sec). Leaf size: 61

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*cos(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{6} \cos ^{3}{\left (x \right )}}{720} + \frac {x^{4} \cos ^{2}{\left (x \right )}}{24} - \frac {x^{2} \cos {\left (x \right )}}{2} + 1\right ) + C_{1} x \left (\frac {x^{4} \cos ^{2}{\left (x \right )}}{120} - \frac {x^{2} \cos {\left (x \right )}}{6} + 1\right ) + O\left (x^{8}\right ) \]