40.2.2 problem 3

Internal problem ID [8591]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.3. Extended Power Series Method: Frobenius Method page 186
Problem number : 3
Date solved : Tuesday, September 30, 2025 at 05:39:29 PM
CAS classification : [_Lienard]

\begin{align*} x y^{\prime \prime }+2 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: 32

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

\[ y = c_1 \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 42

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

\[ y(x)\to c_1 \left (\frac {x^3}{24}-\frac {x}{2}+\frac {1}{x}\right )+c_2 \left (\frac {x^4}{120}-\frac {x^2}{6}+1\right ) \]

✓ Sympy. Time used: 0.264 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{120} - \frac {x^{2}}{6} + 1\right ) + \frac {C_{1} \left (- \frac {x^{6}}{720} + \frac {x^{4}}{24} - \frac {x^{2}}{2} + 1\right )}{x} + O\left (x^{6}\right ) \]