73.24.26 problem 34.9 b(ii)

Internal problem ID [15706]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 34. Power series solutions II: Generalization and theory. Additional Exercises. page 678
Problem number : 34.9 b(ii)
Date solved : Tuesday, January 28, 2025 at 08:05:41 AM
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*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 59

Order:=10; 
dsolve(diff(y(x),x$2)+cos(x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{12} x^{4}-\frac {1}{80} x^{6}+\frac {11}{8064} x^{8}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{30} x^{5}-\frac {19}{5040} x^{7}+\frac {29}{72576} x^{9}\right ) y^{\prime }\left (0\right )+O\left (x^{10}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 70

AsymptoticDSolveValue[D[y[x],{x,2}]+Cos[x]*y[x]==0,y[x],{x,0,"10"-1}]
 

\[ y(x)\to c_2 \left (\frac {29 x^9}{72576}-\frac {19 x^7}{5040}+\frac {x^5}{30}-\frac {x^3}{6}+x\right )+c_1 \left (\frac {11 x^8}{8064}-\frac {x^6}{80}+\frac {x^4}{12}-\frac {x^2}{2}+1\right ) \]