64.15.10 problem 10

Internal problem ID [13587]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 10
Date solved : Tuesday, January 28, 2025 at 05:53:25 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} 2 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: 0.042 (sec). Leaf size: 44

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

\[ y = c_{1} \sqrt {x}\, \left (1-\frac {2}{3} x +\frac {2}{15} x^{2}-\frac {4}{315} x^{3}+\frac {2}{2835} x^{4}-\frac {4}{155925} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-2 x +\frac {2}{3} x^{2}-\frac {4}{45} x^{3}+\frac {2}{315} x^{4}-\frac {4}{14175} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 83

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {4 x^5}{155925}+\frac {2 x^4}{2835}-\frac {4 x^3}{315}+\frac {2 x^2}{15}-\frac {2 x}{3}+1\right )+c_2 \left (-\frac {4 x^5}{14175}+\frac {2 x^4}{315}-\frac {4 x^3}{45}+\frac {2 x^2}{3}-2 x+1\right ) \]