20.28.4 problem 4

Internal problem ID [4066]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Additional problems. Section 11.7. page 788
Problem number : 4
Date solved : Monday, January 27, 2025 at 08:07:24 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 44

Order:=6; 
dsolve(x*diff(y(x),x$2)+diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-2 x +x^{2}-\frac {2}{9} x^{3}+\frac {1}{36} x^{4}-\frac {1}{450} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (4 x -3 x^{2}+\frac {22}{27} x^{3}-\frac {25}{216} x^{4}+\frac {137}{13500} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 101

AsymptoticDSolveValue[x*D[y[x],{x,2}]+D[y[x],x]+2*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {x^5}{450}+\frac {x^4}{36}-\frac {2 x^3}{9}+x^2-2 x+1\right )+c_2 \left (\frac {137 x^5}{13500}-\frac {25 x^4}{216}+\frac {22 x^3}{27}-3 x^2+\left (-\frac {x^5}{450}+\frac {x^4}{36}-\frac {2 x^3}{9}+x^2-2 x+1\right ) \log (x)+4 x\right ) \]