64.15.23 problem 23

Internal problem ID [13529]
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 : 23
Date solved : Monday, March 31, 2025 at 08:00:31 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*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 44

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

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-2 x +x^{2}-\frac {2}{9} x^{3}+\frac {1}{36} x^{4}-\frac {1}{450} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (4 x -3 x^{2}+\frac {22}{27} x^{3}-\frac {25}{216} x^{4}+\frac {137}{13500} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 101

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

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

✓ Sympy. Time used: 0.708 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(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)
 

\[ y{\left (x \right )} = C_{1} \left (- \frac {x^{5}}{450} + \frac {x^{4}}{36} - \frac {2 x^{3}}{9} + x^{2} - 2 x + 1\right ) + O\left (x^{6}\right ) \]