52.1.27 problem 25 expansion at 0

Internal problem ID [8244]
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 : 25 expansion at 0
Date solved : Monday, January 27, 2025 at 03:46:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \cos \left (x \right ) y^{\prime \prime }+y^{\prime }+5 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 74

Order:=8; 
dsolve(cos(x)*diff(y(x),x$2)+diff(y(x),x)+5*y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {5}{2} x^{2}+\frac {5}{6} x^{3}+\frac {5}{8} x^{4}-\frac {5}{24} x^{5}+\frac {1}{16} x^{6}-\frac {13}{336} x^{7}\right ) y \left (0\right )+\left (x -\frac {1}{2} x^{2}-\frac {2}{3} x^{3}+\frac {1}{3} x^{4}+\frac {1}{80} x^{6}+\frac {11}{5040} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 91

AsymptoticDSolveValue[Cos[x]*D[y[x],{x,2}]+D[y[x],x]+5*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_2 \left (\frac {11 x^7}{5040}+\frac {x^6}{80}+\frac {x^4}{3}-\frac {2 x^3}{3}-\frac {x^2}{2}+x\right )+c_1 \left (-\frac {13 x^7}{336}+\frac {x^6}{16}-\frac {5 x^5}{24}+\frac {5 x^4}{8}+\frac {5 x^3}{6}-\frac {5 x^2}{2}+1\right ) \]