36.5.2 problem 2

Internal problem ID [6346]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number : 2
Date solved : Sunday, March 30, 2025 at 10:53:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+3 y^{\prime }-x y&=0 \end{align*}

Using series method with expansion around

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

✗ Maple

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+3*diff(y(x),x)-x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 85

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

\[ y(x)\to c_2 e^{3/x} \left (\frac {3001 x^5}{1620}+\frac {613 x^4}{648}+\frac {16 x^3}{27}+\frac {x^2}{2}+\frac {2 x}{3}+1\right ) x^2+c_1 \left (-\frac {23 x^5}{810}+\frac {7 x^4}{216}-\frac {x^3}{27}+\frac {x^2}{6}+1\right ) \]

✗ Sympy

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

ValueError : ODE x**2*Derivative(y(x), (x, 2)) - x*y(x) + 3*Derivative(y(x), x) does not match hint 2nd_power_series_regular