84.36.2 problem 21.25

Internal problem ID [22342]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 21. Gamma function. Bessel function. Supplementary problems
Problem number : 21.25
Date solved : Thursday, October 02, 2025 at 08:37:47 PM
CAS classification : [_Lienard]

\begin{align*} x y^{\prime \prime }-3 y^{\prime }+y x&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 40

Order:=6; 
ode:=x*diff(diff(y(x),x),x)-3*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{4} \left (1-\frac {1}{12} x^{2}+\frac {1}{384} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (9 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-36 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

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

ode=x*D[y[x],{x,2}]-3*D[y[x],x]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {1}{64} \left (x^2+8\right )^2-\frac {1}{16} x^4 \log (x)\right )+c_2 \left (\frac {x^8}{384}-\frac {x^6}{12}+x^4\right ) \]

✓ Sympy. Time used: 0.219 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) + x*Derivative(y(x), (x, 2)) - 3*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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