Internal problem ID [5827]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+y \sin \relax (x )=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 44
Order:=8; dsolve(x*diff(y(x),x$2)+sin(x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{18} x^{4}-\frac {53}{10800} x^{6}\right ) y \relax (0)+\left (x -\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {19}{15120} x^{7}\right ) D\relax (y )\relax (0)+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 56
AsymptoticDSolveValue[x*y''[x]+Sin[x]*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (-\frac {19 x^7}{15120}+\frac {x^5}{60}-\frac {x^3}{6}+x\right )+c_1 \left (-\frac {53 x^6}{10800}+\frac {x^4}{18}-\frac {x^2}{2}+1\right ) \]