28.5.8 problem 9.8

Internal problem ID [4595]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 9. Series Solutions of Differential Equations. Problems at page 426
Problem number : 9.8
Date solved : Monday, January 27, 2025 at 09:25:53 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (1+\cos \left (x \right )\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: 39

Order:=6; 
dsolve(diff(y(x),x$2)+(1+cos(x))*y(x)=0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \left (1-x^{2}+\frac {5}{24} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{3} x^{3}+\frac {7}{120} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 40

AsymptoticDSolveValue[D[y[x],{x,2}]+(1+Cos[x])*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {7 x^5}{120}-\frac {x^3}{3}+x\right )+c_1 \left (\frac {5 x^4}{24}-x^2+1\right ) \]