Internal problem ID [5895]
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. CHAPTER 6 IN REVIEW.
Page 271
Problem number: 14.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\cos \relax (x ) y^{\prime \prime }+y=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: 39
Order:=8; dsolve(cos(x)*diff(y(x),x$2)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1-\frac {1}{2} x^{2}+\frac {1}{720} x^{6}\right ) y \relax (0)+\left (x -\frac {1}{6} x^{3}-\frac {1}{60} x^{5}-\frac {13}{5040} x^{7}\right ) D\relax (y )\relax (0)+O\left (x^{8}\right ) \]
✓ Solution by Mathematica
Time used: 0.001 (sec). Leaf size: 49
AsymptoticDSolveValue[Cos[x]*y''[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {x^6}{720}-\frac {x^2}{2}+1\right )+c_2 \left (-\frac {13 x^7}{5040}-\frac {x^5}{60}-\frac {x^3}{6}+x\right ) \]