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 : Monday, January 27, 2025 at 03:47:03 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*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 24

Order:=8; 
dsolve([diff(y(x),x$2)+cos(x)*y(x)=0,y(0) = 1, D(y)(0) = 1],y(x),type='series',x=0);
 

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

✓ Solution by Mathematica

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

AsymptoticDSolveValue[{D[y[x],{x,2}]+Cos[x]*y[x]==0,{y[0]==1,Derivative[1][y][0] ==1}},y[x],{x,0,"8"-1}]
 

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