54.3.10 problem 10

Internal problem ID [8566]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 17. Power series solutions. 17.5. Solutions Near an Ordinary Point. Exercises page 355
Problem number : 10
Date solved : Monday, January 27, 2025 at 04:16:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 x y^{\prime }+5 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 49

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

\[ y = \left (1-\frac {5}{2} x^{2}+\frac {15}{8} x^{4}-\frac {13}{16} x^{6}\right ) y \left (0\right )+\left (x -\frac {7}{6} x^{3}+\frac {77}{120} x^{5}-\frac {11}{48} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 56

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

\[ y(x)\to c_2 \left (-\frac {11 x^7}{48}+\frac {77 x^5}{120}-\frac {7 x^3}{6}+x\right )+c_1 \left (-\frac {13 x^6}{16}+\frac {15 x^4}{8}-\frac {5 x^2}{2}+1\right ) \]