74.17.9 problem 9

Internal problem ID [16461]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number : 9
Date solved : Thursday, March 13, 2025 at 08:14:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x -1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.072 (sec). Leaf size: 62

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(x-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} x^{2} \left (1-\frac {1}{3} x +\frac {1}{24} x^{2}-\frac {1}{360} x^{3}+\frac {1}{8640} x^{4}-\frac {1}{302400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{3} x^{3}+\frac {1}{24} x^{4}-\frac {1}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-2 x +\frac {4}{9} x^{3}-\frac {25}{288} x^{4}+\frac {157}{21600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 83

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

\[ y(x)\to c_1 \left (\frac {31 x^4-176 x^3+144 x^2+576 x+576}{576 x}-\frac {1}{48} x \left (x^2-8 x+24\right ) \log (x)\right )+c_2 \left (\frac {x^5}{8640}-\frac {x^4}{360}+\frac {x^3}{24}-\frac {x^2}{3}+x\right ) \]

✓ Sympy. Time used: 0.780 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (x - 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {x^{4}}{8640} - \frac {x^{3}}{360} + \frac {x^{2}}{24} - \frac {x}{3} + 1\right ) + O\left (x^{6}\right ) \]