43.2.3 problem 7.3.5

Internal problem ID [6860]
Book : Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section : Chapter 7. POWER SERIES METHODS. 7.3.2 The method of Frobenius. Exercises. page 300
Problem number : 7.3.5
Date solved : Sunday, March 30, 2025 at 11:24:40 AM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 28

Order:=6; 
ode:=diff(diff(y(x),x),x)+1/x*diff(y(x),x)-x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+\frac {1}{9} x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\left (-\frac {2}{27} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 39

ode=D[y[x],{x,2}]+1/x*D[y[x],x]-x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^3}{9}+1\right )+c_2 \left (\left (\frac {x^3}{9}+1\right ) \log (x)-\frac {2 x^3}{27}\right ) \]

✓ Sympy. Time used: 0.923 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{3}}{9} + 1\right ) + O\left (x^{6}\right ) \]