64.15.10 problem 10

Internal problem ID [13516]
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 : Monday, March 31, 2025 at 08:00:08 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*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 44

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

\[ y = c_1 \sqrt {x}\, \left (1-\frac {2}{3} x +\frac {2}{15} x^{2}-\frac {4}{315} x^{3}+\frac {2}{2835} x^{4}-\frac {4}{155925} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-2 x +\frac {2}{3} x^{2}-\frac {4}{45} x^{3}+\frac {2}{315} x^{4}-\frac {4}{14175} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

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

ode=2*x*D[y[x],{x,2}]+D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.806 (sec). Leaf size: 73

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), (x, 2)) + 2*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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