50.22.11 problem 2(c)

Internal problem ID [8154]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Problems for review and discovert. (A) Drill Exercises . Page 194
Problem number : 2(c)
Date solved : Sunday, March 30, 2025 at 12:46:52 PM
CAS classification : [_Lienard]

\begin{align*} x y^{\prime \prime }-4 y^{\prime }+x y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 36

Order:=8; 
ode:=x*diff(diff(y(x),x),x)-4*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{5} \left (1-\frac {1}{14} x^{2}+\frac {1}{504} x^{4}-\frac {1}{33264} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (2880+480 x^{2}+120 x^{4}-20 x^{6}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 58

ode=x*D[y[x],{x,2}]-4*D[y[x],x]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (-\frac {x^6}{144}+\frac {x^4}{24}+\frac {x^2}{6}+1\right )+c_2 \left (-\frac {x^{11}}{33264}+\frac {x^9}{504}-\frac {x^7}{14}+x^5\right ) \]

✓ Sympy. Time used: 0.793 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) + x*Derivative(y(x), (x, 2)) - 4*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{6}}{144} + \frac {x^{4}}{24} + \frac {x^{2}}{6} + 1\right ) + C_{1} x^{5} \left (1 - \frac {x^{2}}{14}\right ) + O\left (x^{8}\right ) \]