52.1.31 problem 27

Internal problem ID [8248]
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 : 27
Date solved : Monday, January 27, 2025 at 03:46:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\sin \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: 49

Order:=8; 
dsolve(x*diff(y(x),x$2)+sin(x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{2} x^{2}+\frac {1}{18} x^{4}-\frac {53}{10800} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {19}{15120} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 56

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

\[ y(x)\to c_2 \left (-\frac {19 x^7}{15120}+\frac {x^5}{60}-\frac {x^3}{6}+x\right )+c_1 \left (-\frac {53 x^6}{10800}+\frac {x^4}{18}-\frac {x^2}{2}+1\right ) \]