44.22.4 problem 1(d)

Internal problem ID [9433]
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 : 1(d)
Date solved : Tuesday, September 30, 2025 at 06:18:45 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 49

Order:=8; 
ode:=2*diff(diff(y(x),x),x)+x*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{4} x^{2}+\frac {1}{32} x^{4}-\frac {1}{384} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {1}{60} x^{5}-\frac {1}{840} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 56

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

\[ y(x)\to c_2 \left (-\frac {x^7}{840}+\frac {x^5}{60}-\frac {x^3}{6}+x\right )+c_1 \left (-\frac {x^6}{384}+\frac {x^4}{32}-\frac {x^2}{4}+1\right ) \]

✓ Sympy. Time used: 0.210 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {x^{6}}{384} + \frac {x^{4}}{32} - \frac {x^{2}}{4} + 1\right ) + C_{1} x \left (\frac {x^{4}}{60} - \frac {x^{2}}{6} + 1\right ) + O\left (x^{8}\right ) \]