64.15.24 problem 24

Internal problem ID [13522]
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 : 24
Date solved : Wednesday, March 05, 2025 at 10:03:08 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 2 x y^{\prime \prime }+6 y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.055 (sec). Leaf size: 62

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

\[ y = \frac {c_{1} \left (1-\frac {1}{6} x +\frac {1}{96} x^{2}-\frac {1}{2880} x^{3}+\frac {1}{138240} x^{4}-\frac {1}{9676800} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x^{2}+c_{2} \left (\ln \left (x \right ) \left (\frac {1}{4} x^{2}-\frac {1}{24} x^{3}+\frac {1}{384} x^{4}-\frac {1}{11520} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-x +\frac {1}{18} x^{3}-\frac {25}{4608} x^{4}+\frac {157}{691200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 80

ode=2*x*D[y[x],{x,2}]+6*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^4}{138240}-\frac {x^3}{2880}+\frac {x^2}{96}-\frac {x}{6}+1\right )+c_1 \left (\frac {31 x^4-352 x^3+576 x^2+4608 x+9216}{9216 x^2}-\frac {1}{768} \left (x^2-16 x+96\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.727 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = C_{1} \left (- \frac {x^{5}}{9676800} + \frac {x^{4}}{138240} - \frac {x^{3}}{2880} + \frac {x^{2}}{96} - \frac {x}{6} + 1\right ) + O\left (x^{6}\right ) \]