46.2.7 problem 8

Internal problem ID [7310]
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 : 8
Date solved : Wednesday, March 05, 2025 at 04:23:05 AM
CAS classification : [_Lienard]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 32

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

\[ y = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 60

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

\[ y(x)\to c_1 \left (\frac {x^4}{64}-\frac {x^2}{4}+1\right )+c_2 \left (-\frac {3 x^4}{128}+\frac {x^2}{4}+\left (\frac {x^4}{64}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.843 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{4}}{64} - \frac {x^{2}}{4} + 1\right ) + O\left (x^{6}\right ) \]